Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 17.0s

Alternatives: 10

Speedup: 1.0×

• 49.8% 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

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

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(sin(y) / Float64(y / cosh(x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-/r/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \frac{\sin y}{\frac{y}{\cosh x}} \]

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 72.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00036:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\cosh x}{y}}{\frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00036)
   (/ (sin y) y)
   (if (<= x 2.8e+152)
     (/ (/ (cosh x) y) (/ 1.0 y))
     (* 0.5 (/ (sin y) (/ y (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00036) {
		tmp = sin(y) / y;
	} else if (x <= 2.8e+152) {
		tmp = (cosh(x) / y) / (1.0 / y);
	} else {
		tmp = 0.5 * (sin(y) / (y / (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.00036d0) then
        tmp = sin(y) / y
    else if (x <= 2.8d+152) then
        tmp = (cosh(x) / y) / (1.0d0 / y)
    else
        tmp = 0.5d0 * (sin(y) / (y / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.00036) {
		tmp = Math.sin(y) / y;
	} else if (x <= 2.8e+152) {
		tmp = (Math.cosh(x) / y) / (1.0 / y);
	} else {
		tmp = 0.5 * (Math.sin(y) / (y / (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00036:
		tmp = math.sin(y) / y
	elif x <= 2.8e+152:
		tmp = (math.cosh(x) / y) / (1.0 / y)
	else:
		tmp = 0.5 * (math.sin(y) / (y / (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00036)
		tmp = Float64(sin(y) / y);
	elseif (x <= 2.8e+152)
		tmp = Float64(Float64(cosh(x) / y) / Float64(1.0 / y));
	else
		tmp = Float64(0.5 * Float64(sin(y) / Float64(y / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.00036)
		tmp = sin(y) / y;
	elseif (x <= 2.8e+152)
		tmp = (cosh(x) / y) / (1.0 / y);
	else
		tmp = 0.5 * (sin(y) / (y / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00036], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 2.8e+152], N[(N[(N[Cosh[x], $MachinePrecision] / y), $MachinePrecision] / N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sin[y], $MachinePrecision] / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.60000000000000023e-4

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 68.7%

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

   if 3.60000000000000023e-4 < x < 2.8000000000000002e152

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. associate-*l/ 100.0%

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

      3. add-sqr-sqrt 42.5%

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

      4. times-frac 42.5%

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

      5. pow1/2 95.0%

         \[\leadsto \frac{1}{\color{blue}{{\left(\frac{y}{\cosh x}\right)}^{0.5}}} \cdot \frac{\sin y}{\sqrt{\frac{y}{\cosh x}}} \]

      6. pow-flip 95.0%

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

      7. metadata-eval 95.0%

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

      8. associate-*r/ 95.0%

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

      9. associate-/l* 95.0%

         \[\leadsto \color{blue}{\frac{{\left(\frac{y}{\cosh x}\right)}^{-0.5}}{\frac{\sqrt{\frac{y}{\cosh x}}}{\sin y}}} \]

      10. metadata-eval 95.0%

          \[\leadsto \frac{{\left(\frac{y}{\cosh x}\right)}^{\color{blue}{\left(-0.5\right)}}}{\frac{\sqrt{\frac{y}{\cosh x}}}{\sin y}} \]

      11. pow-flip 95.0%

          \[\leadsto \frac{\color{blue}{\frac{1}{{\left(\frac{y}{\cosh x}\right)}^{0.5}}}}{\frac{\sqrt{\frac{y}{\cosh x}}}{\sin y}} \]

      12. pow1/2 42.5%

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

      13. metadata-eval 42.5%

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

      14. sqrt-div 42.5%

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

      15. clear-num 42.5%

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

   7. Applied egg-rr 42.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{\cosh x}{y}}}{\frac{\sqrt{\frac{y}{\cosh x}}}{\sin y}}} \]
   8. Step-by-step derivation

      1. div-inv 42.5%

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

      2. clear-num 42.5%

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

      3. div-inv 42.5%

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

      4. metadata-eval 42.5%

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

      5. sqrt-div 42.5%

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

      6. clear-num 42.5%

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

   9. Applied egg-rr 42.5%

      \[\leadsto \color{blue}{\sqrt{\frac{\cosh x}{y}} \cdot \left(\sin y \cdot \sqrt{\frac{\cosh x}{y}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 42.5%

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

       2. /-rgt-identity 42.5%

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

       3. associate-/r/ 42.5%

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

       4. associate-*l/ 42.5%

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

       5. rem-square-sqrt 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in y around 0 82.5%

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

   if 2.8000000000000002e152 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 97.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\sin y \cdot {x}^{2}}{y}} \]
   10. Step-by-step derivation

       1. unpow2 97.0%

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

       2. associate-/l* 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00036:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\cosh x}{y}}{\frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \end{array} \]

Alternative 4: 84.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00036:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(x \cdot x\right)}{\frac{y}{\sin y}}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\cosh x}{y}}{\frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00036)
   (/ (+ 1.0 (* 0.5 (* x x))) (/ y (sin y)))
   (if (<= x 2.8e+152)
     (/ (/ (cosh x) y) (/ 1.0 y))
     (* 0.5 (/ (sin y) (/ y (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00036) {
		tmp = (1.0 + (0.5 * (x * x))) / (y / sin(y));
	} else if (x <= 2.8e+152) {
		tmp = (cosh(x) / y) / (1.0 / y);
	} else {
		tmp = 0.5 * (sin(y) / (y / (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.00036d0) then
        tmp = (1.0d0 + (0.5d0 * (x * x))) / (y / sin(y))
    else if (x <= 2.8d+152) then
        tmp = (cosh(x) / y) / (1.0d0 / y)
    else
        tmp = 0.5d0 * (sin(y) / (y / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.00036) {
		tmp = (1.0 + (0.5 * (x * x))) / (y / Math.sin(y));
	} else if (x <= 2.8e+152) {
		tmp = (Math.cosh(x) / y) / (1.0 / y);
	} else {
		tmp = 0.5 * (Math.sin(y) / (y / (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00036:
		tmp = (1.0 + (0.5 * (x * x))) / (y / math.sin(y))
	elif x <= 2.8e+152:
		tmp = (math.cosh(x) / y) / (1.0 / y)
	else:
		tmp = 0.5 * (math.sin(y) / (y / (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00036)
		tmp = Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) / Float64(y / sin(y)));
	elseif (x <= 2.8e+152)
		tmp = Float64(Float64(cosh(x) / y) / Float64(1.0 / y));
	else
		tmp = Float64(0.5 * Float64(sin(y) / Float64(y / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.00036)
		tmp = (1.0 + (0.5 * (x * x))) / (y / sin(y));
	elseif (x <= 2.8e+152)
		tmp = (cosh(x) / y) / (1.0 / y);
	else
		tmp = 0.5 * (sin(y) / (y / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00036], N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e+152], N[(N[(N[Cosh[x], $MachinePrecision] / y), $MachinePrecision] / N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sin[y], $MachinePrecision] / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.60000000000000023e-4

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.7%

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

      3. clear-num 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 89.8%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 89.8%

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

   8. Simplified 89.8%

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

   9. Taylor expanded in y around inf 86.4%

      \[\leadsto \color{blue}{\frac{\left(1 + 0.5 \cdot {x}^{2}\right) \cdot \sin y}{y}} \]
   10. Step-by-step derivation

       1. associate-/l* 86.3%

          \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot {x}^{2}}{\frac{y}{\sin y}}} \]

       2. unpow2 86.3%

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

   11. Simplified 86.3%

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

   if 3.60000000000000023e-4 < x < 2.8000000000000002e152

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. associate-*l/ 100.0%

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

      3. add-sqr-sqrt 42.5%

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

      4. times-frac 42.5%

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

      5. pow1/2 95.0%

         \[\leadsto \frac{1}{\color{blue}{{\left(\frac{y}{\cosh x}\right)}^{0.5}}} \cdot \frac{\sin y}{\sqrt{\frac{y}{\cosh x}}} \]

      6. pow-flip 95.0%

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

      7. metadata-eval 95.0%

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

      8. associate-*r/ 95.0%

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

      9. associate-/l* 95.0%

         \[\leadsto \color{blue}{\frac{{\left(\frac{y}{\cosh x}\right)}^{-0.5}}{\frac{\sqrt{\frac{y}{\cosh x}}}{\sin y}}} \]

      10. metadata-eval 95.0%

          \[\leadsto \frac{{\left(\frac{y}{\cosh x}\right)}^{\color{blue}{\left(-0.5\right)}}}{\frac{\sqrt{\frac{y}{\cosh x}}}{\sin y}} \]

      11. pow-flip 95.0%

          \[\leadsto \frac{\color{blue}{\frac{1}{{\left(\frac{y}{\cosh x}\right)}^{0.5}}}}{\frac{\sqrt{\frac{y}{\cosh x}}}{\sin y}} \]

      12. pow1/2 42.5%

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

      13. metadata-eval 42.5%

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

      14. sqrt-div 42.5%

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

      15. clear-num 42.5%

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

   7. Applied egg-rr 42.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{\cosh x}{y}}}{\frac{\sqrt{\frac{y}{\cosh x}}}{\sin y}}} \]
   8. Step-by-step derivation

      1. div-inv 42.5%

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

      2. clear-num 42.5%

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

      3. div-inv 42.5%

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

      4. metadata-eval 42.5%

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

      5. sqrt-div 42.5%

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

      6. clear-num 42.5%

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

   9. Applied egg-rr 42.5%

      \[\leadsto \color{blue}{\sqrt{\frac{\cosh x}{y}} \cdot \left(\sin y \cdot \sqrt{\frac{\cosh x}{y}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 42.5%

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

       2. /-rgt-identity 42.5%

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

       3. associate-/r/ 42.5%

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

       4. associate-*l/ 42.5%

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

       5. rem-square-sqrt 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in y around 0 82.5%

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

   if 2.8000000000000002e152 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 97.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\sin y \cdot {x}^{2}}{y}} \]
   10. Step-by-step derivation

       1. unpow2 97.0%

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

       2. associate-/l* 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00036:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(x \cdot x\right)}{\frac{y}{\sin y}}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\cosh x}{y}}{\frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \end{array} \]

Alternative 5: 68.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000205:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x}{y}}{\frac{1}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.000205) (/ (sin y) y) (/ (/ (cosh x) y) (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.000205) {
		tmp = sin(y) / y;
	} else {
		tmp = (cosh(x) / y) / (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.000205d0) then
        tmp = sin(y) / y
    else
        tmp = (cosh(x) / y) / (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.000205) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = (Math.cosh(x) / y) / (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.000205:
		tmp = math.sin(y) / y
	else:
		tmp = (math.cosh(x) / y) / (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.000205)
		tmp = Float64(sin(y) / y);
	else
		tmp = Float64(Float64(cosh(x) / y) / Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.000205)
		tmp = sin(y) / y;
	else
		tmp = (cosh(x) / y) / (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.000205], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[(N[(N[Cosh[x], $MachinePrecision] / y), $MachinePrecision] / N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.05e-4

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 68.7%

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

   if 2.05e-4 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. associate-*l/ 100.0%

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

      3. add-sqr-sqrt 42.0%

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

      4. times-frac 42.0%

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

      5. pow1/2 97.1%

         \[\leadsto \frac{1}{\color{blue}{{\left(\frac{y}{\cosh x}\right)}^{0.5}}} \cdot \frac{\sin y}{\sqrt{\frac{y}{\cosh x}}} \]

      6. pow-flip 97.1%

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

      7. metadata-eval 97.1%

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

      8. associate-*r/ 97.1%

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

      9. associate-/l* 97.1%

         \[\leadsto \color{blue}{\frac{{\left(\frac{y}{\cosh x}\right)}^{-0.5}}{\frac{\sqrt{\frac{y}{\cosh x}}}{\sin y}}} \]

      10. metadata-eval 97.1%

          \[\leadsto \frac{{\left(\frac{y}{\cosh x}\right)}^{\color{blue}{\left(-0.5\right)}}}{\frac{\sqrt{\frac{y}{\cosh x}}}{\sin y}} \]

      11. pow-flip 97.1%

          \[\leadsto \frac{\color{blue}{\frac{1}{{\left(\frac{y}{\cosh x}\right)}^{0.5}}}}{\frac{\sqrt{\frac{y}{\cosh x}}}{\sin y}} \]

      12. pow1/2 42.0%

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

      13. metadata-eval 42.0%

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

      14. sqrt-div 42.0%

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

      15. clear-num 42.0%

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

   7. Applied egg-rr 42.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{\cosh x}{y}}}{\frac{\sqrt{\frac{y}{\cosh x}}}{\sin y}}} \]
   8. Step-by-step derivation

      1. div-inv 42.0%

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

      2. clear-num 42.0%

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

      3. div-inv 42.0%

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

      4. metadata-eval 42.0%

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

      5. sqrt-div 42.0%

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

      6. clear-num 42.0%

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

   9. Applied egg-rr 42.0%

      \[\leadsto \color{blue}{\sqrt{\frac{\cosh x}{y}} \cdot \left(\sin y \cdot \sqrt{\frac{\cosh x}{y}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 42.0%

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

       2. /-rgt-identity 42.0%

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

       3. associate-/r/ 42.0%

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

       4. associate-*l/ 42.0%

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

       5. rem-square-sqrt 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in y around 0 79.7%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000205:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x}{y}}{\frac{1}{y}}\\ \end{array} \]

Alternative 6: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 11:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+173} \lor \neg \left(x \leq 6.4 \cdot 10^{+254}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 11.0)
   (/ (sin y) y)
   (if (or (<= x 2e+173) (not (<= x 6.4e+254)))
     (* (* x x) (+ 0.5 (* -0.08333333333333333 (* y y))))
     (+ 1.0 (* 0.5 (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 11.0) {
		tmp = sin(y) / y;
	} else if ((x <= 2e+173) || !(x <= 6.4e+254)) {
		tmp = (x * x) * (0.5 + (-0.08333333333333333 * (y * y)));
	} else {
		tmp = 1.0 + (0.5 * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 11.0d0) then
        tmp = sin(y) / y
    else if ((x <= 2d+173) .or. (.not. (x <= 6.4d+254))) then
        tmp = (x * x) * (0.5d0 + ((-0.08333333333333333d0) * (y * y)))
    else
        tmp = 1.0d0 + (0.5d0 * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 11.0) {
		tmp = Math.sin(y) / y;
	} else if ((x <= 2e+173) || !(x <= 6.4e+254)) {
		tmp = (x * x) * (0.5 + (-0.08333333333333333 * (y * y)));
	} else {
		tmp = 1.0 + (0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 11.0:
		tmp = math.sin(y) / y
	elif (x <= 2e+173) or not (x <= 6.4e+254):
		tmp = (x * x) * (0.5 + (-0.08333333333333333 * (y * y)))
	else:
		tmp = 1.0 + (0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 11.0)
		tmp = Float64(sin(y) / y);
	elseif ((x <= 2e+173) || !(x <= 6.4e+254))
		tmp = Float64(Float64(x * x) * Float64(0.5 + Float64(-0.08333333333333333 * Float64(y * y))));
	else
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 11.0)
		tmp = sin(y) / y;
	elseif ((x <= 2e+173) || ~((x <= 6.4e+254)))
		tmp = (x * x) * (0.5 + (-0.08333333333333333 * (y * y)));
	else
		tmp = 1.0 + (0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 11.0], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[Or[LessEqual[x, 2e+173], N[Not[LessEqual[x, 6.4e+254]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(-0.08333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 11

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 68.5%

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

   if 11 < x < 2e173 or 6.3999999999999996e254 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 42.1%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 42.1%

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

   8. Simplified 42.1%

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

   9. Taylor expanded in y around 0 13.7%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\left(1 + 0.5 \cdot {x}^{2}\right) \cdot {y}^{2}\right) + \left(1 + 0.5 \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. fma-def 13.7%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \left(1 + 0.5 \cdot {x}^{2}\right) \cdot {y}^{2}, 1 + 0.5 \cdot {x}^{2}\right)} \]

       2. *-commutative 13.7%

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{y}^{2} \cdot \left(1 + 0.5 \cdot {x}^{2}\right)}, 1 + 0.5 \cdot {x}^{2}\right) \]

       3. unpow2 13.7%

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\left(y \cdot y\right)} \cdot \left(1 + 0.5 \cdot {x}^{2}\right), 1 + 0.5 \cdot {x}^{2}\right) \]

       4. unpow2 13.7%

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \left(y \cdot y\right) \cdot \left(1 + 0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right), 1 + 0.5 \cdot {x}^{2}\right) \]

       5. unpow2 13.7%

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

   11. Simplified 13.7%

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

   12. Taylor expanded in x around inf 34.5%

       \[\leadsto \color{blue}{\left(0.5 + -0.08333333333333333 \cdot {y}^{2}\right) \cdot {x}^{2}} \]
   13. Step-by-step derivation

       1. *-commutative 34.5%

          \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.5 + -0.08333333333333333 \cdot {y}^{2}\right)} \]

       2. unpow2 34.5%

          \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + -0.08333333333333333 \cdot {y}^{2}\right) \]

       3. unpow2 34.5%

          \[\leadsto \left(x \cdot x\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   14. Simplified 34.5%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(y \cdot y\right)\right)} \]

   if 2e173 < x < 6.3999999999999996e254

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in y around 0 78.6%

      \[\leadsto \color{blue}{1 + 0.5 \cdot {x}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 78.6%

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

   11. Simplified 78.6%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 11:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+173} \lor \neg \left(x \leq 6.4 \cdot 10^{+254}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 7: 44.7% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;y \leq 5.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{t_0}{1 + \left(y \cdot y\right) \cdot 0.16666666666666666}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+201}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.5 (* x x)))))
   (if (<= y 5.2e+47)
     (/ t_0 (+ 1.0 (* (* y y) 0.16666666666666666)))
     (if (<= y 7e+201)
       (* (* x x) (+ 0.5 (* -0.08333333333333333 (* y y))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.5 * (x * x));
	double tmp;
	if (y <= 5.2e+47) {
		tmp = t_0 / (1.0 + ((y * y) * 0.16666666666666666));
	} else if (y <= 7e+201) {
		tmp = (x * x) * (0.5 + (-0.08333333333333333 * (y * y)));
	} else {
		tmp = t_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 = 1.0d0 + (0.5d0 * (x * x))
    if (y <= 5.2d+47) then
        tmp = t_0 / (1.0d0 + ((y * y) * 0.16666666666666666d0))
    else if (y <= 7d+201) then
        tmp = (x * x) * (0.5d0 + ((-0.08333333333333333d0) * (y * y)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.5 * (x * x));
	double tmp;
	if (y <= 5.2e+47) {
		tmp = t_0 / (1.0 + ((y * y) * 0.16666666666666666));
	} else if (y <= 7e+201) {
		tmp = (x * x) * (0.5 + (-0.08333333333333333 * (y * y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.5 * (x * x))
	tmp = 0
	if y <= 5.2e+47:
		tmp = t_0 / (1.0 + ((y * y) * 0.16666666666666666))
	elif y <= 7e+201:
		tmp = (x * x) * (0.5 + (-0.08333333333333333 * (y * y)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.5 * Float64(x * x)))
	tmp = 0.0
	if (y <= 5.2e+47)
		tmp = Float64(t_0 / Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666)));
	elseif (y <= 7e+201)
		tmp = Float64(Float64(x * x) * Float64(0.5 + Float64(-0.08333333333333333 * Float64(y * y))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.5 * (x * x));
	tmp = 0.0;
	if (y <= 5.2e+47)
		tmp = t_0 / (1.0 + ((y * y) * 0.16666666666666666));
	elseif (y <= 7e+201)
		tmp = (x * x) * (0.5 + (-0.08333333333333333 * (y * y)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.2e+47], N[(t$95$0 / N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+201], N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(-0.08333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.20000000000000007e47

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.8%

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

      3. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 82.0%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 82.0%

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

   8. Simplified 82.0%

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

   9. Taylor expanded in y around inf 75.3%

      \[\leadsto \color{blue}{\frac{\left(1 + 0.5 \cdot {x}^{2}\right) \cdot \sin y}{y}} \]
   10. Step-by-step derivation

       1. associate-/l* 75.3%

          \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot {x}^{2}}{\frac{y}{\sin y}}} \]

       2. unpow2 75.3%

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

   11. Simplified 75.3%

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

   12. Taylor expanded in y around 0 51.0%

       \[\leadsto \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   13. Step-by-step derivation

       1. unpow2 51.0%

          \[\leadsto \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}} \]

   14. Simplified 51.0%

       \[\leadsto \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{\color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)}} \]

   if 5.20000000000000007e47 < y < 7.0000000000000004e201

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/r/ 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

      3. clear-num 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 68.4%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 68.4%

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

   8. Simplified 68.4%

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

   9. Taylor expanded in y around 0 14.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\left(1 + 0.5 \cdot {x}^{2}\right) \cdot {y}^{2}\right) + \left(1 + 0.5 \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. fma-def 14.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \left(1 + 0.5 \cdot {x}^{2}\right) \cdot {y}^{2}, 1 + 0.5 \cdot {x}^{2}\right)} \]

       2. *-commutative 14.0%

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{y}^{2} \cdot \left(1 + 0.5 \cdot {x}^{2}\right)}, 1 + 0.5 \cdot {x}^{2}\right) \]

       3. unpow2 14.0%

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\left(y \cdot y\right)} \cdot \left(1 + 0.5 \cdot {x}^{2}\right), 1 + 0.5 \cdot {x}^{2}\right) \]

       4. unpow2 14.0%

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \left(y \cdot y\right) \cdot \left(1 + 0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right), 1 + 0.5 \cdot {x}^{2}\right) \]

       5. unpow2 14.0%

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

   11. Simplified 14.0%

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

   12. Taylor expanded in x around inf 31.9%

       \[\leadsto \color{blue}{\left(0.5 + -0.08333333333333333 \cdot {y}^{2}\right) \cdot {x}^{2}} \]
   13. Step-by-step derivation

       1. *-commutative 31.9%

          \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.5 + -0.08333333333333333 \cdot {y}^{2}\right)} \]

       2. unpow2 31.9%

          \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + -0.08333333333333333 \cdot {y}^{2}\right) \]

       3. unpow2 31.9%

          \[\leadsto \left(x \cdot x\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   14. Simplified 31.9%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(y \cdot y\right)\right)} \]

   if 7.0000000000000004e201 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.7%

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

      3. clear-num 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 94.2%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 94.2%

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

   8. Simplified 94.2%

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

   9. Taylor expanded in y around 0 24.5%

      \[\leadsto \color{blue}{1 + 0.5 \cdot {x}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 24.5%

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

   11. Simplified 24.5%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(x \cdot x\right)}{1 + \left(y \cdot y\right) \cdot 0.16666666666666666}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+201}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 8: 46.1% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+47} \lor \neg \left(y \leq 7 \cdot 10^{+201}\right):\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 5.2e+47) (not (<= y 7e+201)))
   (+ 1.0 (* 0.5 (* x x)))
   (* (* x x) (+ 0.5 (* -0.08333333333333333 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 5.2e+47) || !(y <= 7e+201)) {
		tmp = 1.0 + (0.5 * (x * x));
	} else {
		tmp = (x * x) * (0.5 + (-0.08333333333333333 * (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 5.2d+47) .or. (.not. (y <= 7d+201))) then
        tmp = 1.0d0 + (0.5d0 * (x * x))
    else
        tmp = (x * x) * (0.5d0 + ((-0.08333333333333333d0) * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 5.2e+47) || !(y <= 7e+201)) {
		tmp = 1.0 + (0.5 * (x * x));
	} else {
		tmp = (x * x) * (0.5 + (-0.08333333333333333 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 5.2e+47) or not (y <= 7e+201):
		tmp = 1.0 + (0.5 * (x * x))
	else:
		tmp = (x * x) * (0.5 + (-0.08333333333333333 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 5.2e+47) || !(y <= 7e+201))
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * x)));
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 + Float64(-0.08333333333333333 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 5.2e+47) || ~((y <= 7e+201)))
		tmp = 1.0 + (0.5 * (x * x));
	else
		tmp = (x * x) * (0.5 + (-0.08333333333333333 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 5.2e+47], N[Not[LessEqual[y, 7e+201]], $MachinePrecision]], N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(-0.08333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.20000000000000007e47 or 7.0000000000000004e201 < y

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.8%

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

      3. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 83.1%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 83.1%

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

   8. Simplified 83.1%

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

   9. Taylor expanded in y around 0 50.4%

      \[\leadsto \color{blue}{1 + 0.5 \cdot {x}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 50.4%

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

   11. Simplified 50.4%

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

   if 5.20000000000000007e47 < y < 7.0000000000000004e201

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/r/ 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

      3. clear-num 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 68.4%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 68.4%

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

   8. Simplified 68.4%

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

   9. Taylor expanded in y around 0 14.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\left(1 + 0.5 \cdot {x}^{2}\right) \cdot {y}^{2}\right) + \left(1 + 0.5 \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. fma-def 14.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \left(1 + 0.5 \cdot {x}^{2}\right) \cdot {y}^{2}, 1 + 0.5 \cdot {x}^{2}\right)} \]

       2. *-commutative 14.0%

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{y}^{2} \cdot \left(1 + 0.5 \cdot {x}^{2}\right)}, 1 + 0.5 \cdot {x}^{2}\right) \]

       3. unpow2 14.0%

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\left(y \cdot y\right)} \cdot \left(1 + 0.5 \cdot {x}^{2}\right), 1 + 0.5 \cdot {x}^{2}\right) \]

       4. unpow2 14.0%

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \left(y \cdot y\right) \cdot \left(1 + 0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right), 1 + 0.5 \cdot {x}^{2}\right) \]

       5. unpow2 14.0%

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

   11. Simplified 14.0%

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

   12. Taylor expanded in x around inf 31.9%

       \[\leadsto \color{blue}{\left(0.5 + -0.08333333333333333 \cdot {y}^{2}\right) \cdot {x}^{2}} \]
   13. Step-by-step derivation

       1. *-commutative 31.9%

          \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.5 + -0.08333333333333333 \cdot {y}^{2}\right)} \]

       2. unpow2 31.9%

          \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + -0.08333333333333333 \cdot {y}^{2}\right) \]

       3. unpow2 31.9%

          \[\leadsto \left(x \cdot x\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   14. Simplified 31.9%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+47} \lor \neg \left(y \leq 7 \cdot 10^{+201}\right):\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 9: 45.0% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + 0.5 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* 0.5 (* x x))))

C

double code(double x, double y) {
	return 1.0 + (0.5 * (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (0.5d0 * (x * x))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (0.5 * (x * x));
}

Python

def code(x, y):
	return 1.0 + (0.5 * (x * x))

Julia

function code(x, y)
	return Float64(1.0 + Float64(0.5 * Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (0.5 * (x * x));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-/r/ 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.8%

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

   2. associate-/r/ 99.8%

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

   3. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in x around 0 80.4%

   \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
7. Step-by-step derivation

   1. unpow2 80.4%

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

8. Simplified 80.4%

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

9. Taylor expanded in y around 0 42.6%

   \[\leadsto \color{blue}{1 + 0.5 \cdot {x}^{2}} \]
10. Step-by-step derivation

    1. unpow2 42.6%

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

11. Simplified 42.6%

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

12. Final simplification 42.6%

    \[\leadsto 1 + 0.5 \cdot \left(x \cdot x\right) \]

Alternative 10: 26.8% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-/r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 51.3%

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

5. Taylor expanded in y around 0 24.0%

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

6. Final simplification 24.0%

   \[\leadsto 1 \]

Developer target: 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 2023274 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (/ (* (cosh x) (sin y)) y)

  (* (cosh x) (/ (sin y) y)))