Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 7.5s

Alternatives: 9

Speedup: 1.0×

• 49.1% 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. add-log-exp 73.6%

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

   2. *-un-lft-identity 73.6%

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

   3. log-prod 73.6%

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

   4. metadata-eval 73.6%

      \[\leadsto \color{blue}{0} + \log \left(e^{\cosh x \cdot \frac{\sin y}{y}}\right) \]

   5. add-log-exp 99.9%

      \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\sin y}{y}} \]

   6. *-commutative 99.9%

      \[\leadsto 0 + \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

   7. associate-*l/ 99.9%

      \[\leadsto 0 + \color{blue}{\frac{\sin y \cdot \cosh x}{y}} \]

3. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{0 + \frac{\sin y \cdot \cosh x}{y}} \]
4. Step-by-step derivation

   1. +-lft-identity 99.9%

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

   2. associate-/l* 99.9%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 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. Final simplification 99.9%

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

Alternative 3: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 65:\\ \;\;\;\;\frac{\sin y \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}{y}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+100}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+152}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 65.0)
   (/ (* (sin y) (+ 1.0 (* 0.5 (* x x)))) y)
   (if (<= x 6.5e+100)
     (cosh x)
     (if (<= x 1.75e+152)
       (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))
       (* (sin y) (+ (/ 1.0 y) (* 0.5 (/ (* x x) y))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 65.0) {
		tmp = (sin(y) * (1.0 + (0.5 * (x * x)))) / y;
	} else if (x <= 6.5e+100) {
		tmp = cosh(x);
	} else if (x <= 1.75e+152) {
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = sin(y) * ((1.0 / y) + (0.5 * ((x * x) / 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 <= 65.0d0) then
        tmp = (sin(y) * (1.0d0 + (0.5d0 * (x * x)))) / y
    else if (x <= 6.5d+100) then
        tmp = cosh(x)
    else if (x <= 1.75d+152) then
        tmp = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    else
        tmp = sin(y) * ((1.0d0 / y) + (0.5d0 * ((x * x) / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 65.0) {
		tmp = (Math.sin(y) * (1.0 + (0.5 * (x * x)))) / y;
	} else if (x <= 6.5e+100) {
		tmp = Math.cosh(x);
	} else if (x <= 1.75e+152) {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = Math.sin(y) * ((1.0 / y) + (0.5 * ((x * x) / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 65.0:
		tmp = (math.sin(y) * (1.0 + (0.5 * (x * x)))) / y
	elif x <= 6.5e+100:
		tmp = math.cosh(x)
	elif x <= 1.75e+152:
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = math.sin(y) * ((1.0 / y) + (0.5 * ((x * x) / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 65.0)
		tmp = Float64(Float64(sin(y) * Float64(1.0 + Float64(0.5 * Float64(x * x)))) / y);
	elseif (x <= 6.5e+100)
		tmp = cosh(x);
	elseif (x <= 1.75e+152)
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	else
		tmp = Float64(sin(y) * Float64(Float64(1.0 / y) + Float64(0.5 * Float64(Float64(x * x) / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 65.0)
		tmp = (sin(y) * (1.0 + (0.5 * (x * x)))) / y;
	elseif (x <= 6.5e+100)
		tmp = cosh(x);
	elseif (x <= 1.75e+152)
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = sin(y) * ((1.0 / y) + (0.5 * ((x * x) / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 65.0], N[(N[(N[Sin[y], $MachinePrecision] * N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 6.5e+100], N[Cosh[x], $MachinePrecision], If[LessEqual[x, 1.75e+152], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + N[(0.5 * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 65

   1. Initial program 99.9%

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

      1. add-log-exp 66.8%

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

      2. *-un-lft-identity 66.8%

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

      3. log-prod 66.8%

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

      4. metadata-eval 66.8%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cosh x \cdot \frac{\sin y}{y}}\right) \]

      5. add-log-exp 99.9%

         \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\sin y}{y}} \]

      6. *-commutative 99.9%

         \[\leadsto 0 + \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      7. associate-*l/ 99.9%

         \[\leadsto 0 + \color{blue}{\frac{\sin y \cdot \cosh x}{y}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0 + \frac{\sin y \cdot \cosh x}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 99.9%

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

      2. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

      1. associate-/l* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l/ 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in x around 0 88.9%

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

      1. unpow2 88.9%

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

   10. Simplified 88.9%

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

   11. Taylor expanded in y around inf 86.3%

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

       1. *-commutative 86.3%

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

       2. unpow2 86.3%

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

   13. Simplified 86.3%

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

   if 65 < x < 6.50000000000000001e100

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

   if 6.50000000000000001e100 < x < 1.74999999999999991e152

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 88.9%

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

      1. unpow2 88.9%

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

   4. Simplified 88.9%

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

   if 1.74999999999999991e152 < x

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cosh x \cdot \frac{\sin y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\sin y}{y}} \]

      6. *-commutative 100.0%

         \[\leadsto 0 + \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      7. associate-*l/ 100.0%

         \[\leadsto 0 + \color{blue}{\frac{\sin y \cdot \cosh x}{y}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \frac{\sin y \cdot \cosh x}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

   5. Simplified 100.0%

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

      1. associate-/l* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l/ 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 65:\\ \;\;\;\;\frac{\sin y \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}{y}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+100}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+152}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)\\ \end{array} \]

Alternative 4: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}{y}\\ \mathbf{if}\;x \leq 65:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+98}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin y) (+ 1.0 (* 0.5 (* x x)))) y)))
   (if (<= x 65.0)
     t_0
     (if (<= x 3e+98)
       (cosh x)
       (if (<= x 1.35e+154)
         (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))
         t_0)))))

C

double code(double x, double y) {
	double t_0 = (sin(y) * (1.0 + (0.5 * (x * x)))) / y;
	double tmp;
	if (x <= 65.0) {
		tmp = t_0;
	} else if (x <= 3e+98) {
		tmp = cosh(x);
	} else if (x <= 1.35e+154) {
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (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 = (sin(y) * (1.0d0 + (0.5d0 * (x * x)))) / y
    if (x <= 65.0d0) then
        tmp = t_0
    else if (x <= 3d+98) then
        tmp = cosh(x)
    else if (x <= 1.35d+154) then
        tmp = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (Math.sin(y) * (1.0 + (0.5 * (x * x)))) / y;
	double tmp;
	if (x <= 65.0) {
		tmp = t_0;
	} else if (x <= 3e+98) {
		tmp = Math.cosh(x);
	} else if (x <= 1.35e+154) {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sin(y) * (1.0 + (0.5 * (x * x)))) / y
	tmp = 0
	if x <= 65.0:
		tmp = t_0
	elif x <= 3e+98:
		tmp = math.cosh(x)
	elif x <= 1.35e+154:
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(y) * Float64(1.0 + Float64(0.5 * Float64(x * x)))) / y)
	tmp = 0.0
	if (x <= 65.0)
		tmp = t_0;
	elseif (x <= 3e+98)
		tmp = cosh(x);
	elseif (x <= 1.35e+154)
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sin(y) * (1.0 + (0.5 * (x * x)))) / y;
	tmp = 0.0;
	if (x <= 65.0)
		tmp = t_0;
	elseif (x <= 3e+98)
		tmp = cosh(x);
	elseif (x <= 1.35e+154)
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[y], $MachinePrecision] * N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, 65.0], t$95$0, If[LessEqual[x, 3e+98], N[Cosh[x], $MachinePrecision], If[LessEqual[x, 1.35e+154], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < 65 or 1.35000000000000003e154 < x

   1. Initial program 99.9%

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

      1. add-log-exp 70.8%

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

      2. *-un-lft-identity 70.8%

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

      3. log-prod 70.8%

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

      4. metadata-eval 70.8%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cosh x \cdot \frac{\sin y}{y}}\right) \]

      5. add-log-exp 99.9%

         \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\sin y}{y}} \]

      6. *-commutative 99.9%

         \[\leadsto 0 + \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      7. associate-*l/ 99.9%

         \[\leadsto 0 + \color{blue}{\frac{\sin y \cdot \cosh x}{y}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0 + \frac{\sin y \cdot \cosh x}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 99.9%

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

      2. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

      1. associate-/l* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l/ 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in x around 0 90.2%

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

      1. unpow2 90.2%

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

   10. Simplified 90.2%

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

   11. Taylor expanded in y around inf 88.0%

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

       1. *-commutative 88.0%

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

       2. unpow2 88.0%

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

   13. Simplified 88.0%

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

   if 65 < x < 3.0000000000000001e98

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

   if 3.0000000000000001e98 < x < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 88.9%

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

      1. unpow2 88.9%

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

   4. Simplified 88.9%

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 65:\\ \;\;\;\;\frac{\sin y \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}{y}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+98}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}{y}\\ \end{array} \]

Alternative 5: 68.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+101}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 10^{+260}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8.5e-5)
   (/ (sin y) y)
   (if (<= x 2.3e+101)
     (cosh x)
     (if (<= x 1e+260)
       (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))
       (+ 1.0 (* 0.5 (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8.5e-5) {
		tmp = sin(y) / y;
	} else if (x <= 2.3e+101) {
		tmp = cosh(x);
	} else if (x <= 1e+260) {
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (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 <= 8.5d-5) then
        tmp = sin(y) / y
    else if (x <= 2.3d+101) then
        tmp = cosh(x)
    else if (x <= 1d+260) then
        tmp = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (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 <= 8.5e-5) {
		tmp = Math.sin(y) / y;
	} else if (x <= 2.3e+101) {
		tmp = Math.cosh(x);
	} else if (x <= 1e+260) {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = 1.0 + (0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 8.5e-5:
		tmp = math.sin(y) / y
	elif x <= 2.3e+101:
		tmp = math.cosh(x)
	elif x <= 1e+260:
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = 1.0 + (0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8.5e-5)
		tmp = Float64(sin(y) / y);
	elseif (x <= 2.3e+101)
		tmp = cosh(x);
	elseif (x <= 1e+260)
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(-0.16666666666666666 * 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 <= 8.5e-5)
		tmp = sin(y) / y;
	elseif (x <= 2.3e+101)
		tmp = cosh(x);
	elseif (x <= 1e+260)
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = 1.0 + (0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8.5e-5], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 2.3e+101], N[Cosh[x], $MachinePrecision], If[LessEqual[x, 1e+260], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 8.500000000000001e-5

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 70.7%

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

   if 8.500000000000001e-5 < x < 2.3000000000000001e101

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 89.0%

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

   if 2.3000000000000001e101 < x < 1.00000000000000007e260

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 83.9%

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

      1. unpow2 83.9%

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

   4. Simplified 83.9%

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

   if 1.00000000000000007e260 < x

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cosh x \cdot \frac{\sin y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\sin y}{y}} \]

      6. *-commutative 100.0%

         \[\leadsto 0 + \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      7. associate-*l/ 100.0%

         \[\leadsto 0 + \color{blue}{\frac{\sin y \cdot \cosh x}{y}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \frac{\sin y \cdot \cosh x}{y}} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

   5. Simplified 100.0%

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

      1. associate-/l* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l/ 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in y around 0 100.0%

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

       1. unpow2 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+101}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 10^{+260}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 6: 68.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 8.5e-5) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8.5e-5) {
		tmp = sin(y) / y;
	} else {
		tmp = cosh(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 <= 8.5d-5) then
        tmp = sin(y) / y
    else
        tmp = cosh(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 8.5e-5) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 8.5e-5:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8.5e-5)
		tmp = Float64(sin(y) / y);
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 8.5e-5)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8.5e-5], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.500000000000001e-5

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 70.7%

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

   if 8.500000000000001e-5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 78.0%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 7: 63.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cosh x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (cosh x))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return cosh(x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Cosh[x], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0 62.4%

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

3. Final simplification 62.4%

   \[\leadsto \cosh x \]

Alternative 8: 46.4% 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. add-log-exp 73.6%

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

   2. *-un-lft-identity 73.6%

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

   3. log-prod 73.6%

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

   4. metadata-eval 73.6%

      \[\leadsto \color{blue}{0} + \log \left(e^{\cosh x \cdot \frac{\sin y}{y}}\right) \]

   5. add-log-exp 99.9%

      \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\sin y}{y}} \]

   6. *-commutative 99.9%

      \[\leadsto 0 + \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

   7. associate-*l/ 99.9%

      \[\leadsto 0 + \color{blue}{\frac{\sin y \cdot \cosh x}{y}} \]

3. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{0 + \frac{\sin y \cdot \cosh x}{y}} \]
4. Step-by-step derivation

   1. +-lft-identity 99.9%

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

   2. associate-/l* 99.9%

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

5. Simplified 99.9%

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

   1. associate-/l* 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-*l/ 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in x around 0 84.4%

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

   1. unpow2 84.4%

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

10. Simplified 84.4%

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

11. Taylor expanded in y around 0 47.2%

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

    1. unpow2 47.2%

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

13. Simplified 47.2%

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

14. Final simplification 47.2%

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

Alternative 9: 27.2% 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. add-sqr-sqrt 74.5%

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

   2. pow2 74.5%

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

   3. *-commutative 74.5%

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

   4. associate-*l/ 74.5%

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

3. Applied egg-rr 74.5%

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

4. Taylor expanded in x around 0 40.1%

   \[\leadsto {\left(\sqrt{\frac{\color{blue}{\sin y}}{y}}\right)}^{2} \]

5. Taylor expanded in y around 0 28.4%

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

6. Final simplification 28.4%

   \[\leadsto 1 \]

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 2023238 
(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)))