Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 10.9s

Alternatives: 9

Speedup: 1.0×

• 50.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

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]

Derivation

1. Initial program 99.9%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.0000005:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh x \cdot 4\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.0000005) (/ (sin y) y) (* (* (cosh x) 4.0) 0.25)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.0000005) {
		tmp = sin(y) / y;
	} else {
		tmp = (cosh(x) * 4.0) * 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (cosh(x) <= 1.0000005d0) then
        tmp = sin(y) / y
    else
        tmp = (cosh(x) * 4.0d0) * 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.cosh(x) <= 1.0000005) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = (Math.cosh(x) * 4.0) * 0.25;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.0000005:
		tmp = math.sin(y) / y
	else:
		tmp = (math.cosh(x) * 4.0) * 0.25
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.0000005)
		tmp = Float64(sin(y) / y);
	else
		tmp = Float64(Float64(cosh(x) * 4.0) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (cosh(x) <= 1.0000005)
		tmp = sin(y) / y;
	else
		tmp = (cosh(x) * 4.0) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.0000005], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[(N[(N[Cosh[x], $MachinePrecision] * 4.0), $MachinePrecision] * 0.25), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.0000005000000001

   1. Initial program 99.8%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.1%

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

   if 1.0000005000000001 < (cosh.f64 x)

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

      1. associate-/l* 100.0%

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

      2. cosh-def 100.0%

         \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}}}{\frac{y}{\sin y}} \]

      3. associate-/l/ 100.0%

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

      4. cosh-undef 100.0%

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

      5. cosh-def 100.0%

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

      6. div-inv 100.0%

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

      7. associate-*r* 100.0%

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

      8. times-frac 100.0%

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

      9. cosh-undef 100.0%

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

      10. associate-*r* 100.0%

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

      11. metadata-eval 100.0%

          \[\leadsto \frac{\color{blue}{4} \cdot \cosh x}{\frac{y}{\sin y}} \cdot \frac{\frac{1}{2}}{2} \]

      12. metadata-eval 100.0%

          \[\leadsto \frac{4 \cdot \cosh x}{\frac{y}{\sin y}} \cdot \frac{\color{blue}{0.5}}{2} \]

      13. metadata-eval 100.0%

          \[\leadsto \frac{4 \cdot \cosh x}{\frac{y}{\sin y}} \cdot \color{blue}{0.25} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{4 \cdot \cosh x}{\frac{y}{\sin y}} \cdot 0.25} \]

   6. Taylor expanded in y around 0 73.4%

      \[\leadsto \frac{4 \cdot \cosh x}{\color{blue}{1}} \cdot 0.25 \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.0000005:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh x \cdot 4\right) \cdot 0.25\\ \end{array} \]

Alternative 3: 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 4: 56.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 800:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+181}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{-2 \cdot \left(x \cdot x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 800.0)
   (/ (sin y) y)
   (if (<= x 2.4e+181)
     (+ 1.0 (* (* y y) -0.16666666666666666))
     (+ -2.0 (/ (* -2.0 (* x x)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 800.0) {
		tmp = sin(y) / y;
	} else if (x <= 2.4e+181) {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	} else {
		tmp = -2.0 + ((-2.0 * (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 <= 800.0d0) then
        tmp = sin(y) / y
    else if (x <= 2.4d+181) then
        tmp = 1.0d0 + ((y * y) * (-0.16666666666666666d0))
    else
        tmp = (-2.0d0) + (((-2.0d0) * (x * x)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 800.0) {
		tmp = Math.sin(y) / y;
	} else if (x <= 2.4e+181) {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	} else {
		tmp = -2.0 + ((-2.0 * (x * x)) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 800.0:
		tmp = math.sin(y) / y
	elif x <= 2.4e+181:
		tmp = 1.0 + ((y * y) * -0.16666666666666666)
	else:
		tmp = -2.0 + ((-2.0 * (x * x)) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 800.0)
		tmp = Float64(sin(y) / y);
	elseif (x <= 2.4e+181)
		tmp = Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = Float64(-2.0 + Float64(Float64(-2.0 * Float64(x * x)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 800.0)
		tmp = sin(y) / y;
	elseif (x <= 2.4e+181)
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	else
		tmp = -2.0 + ((-2.0 * (x * x)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 800.0], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 2.4e+181], N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(-2.0 + N[(N[(-2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 800

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 63.7%

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

   if 800 < x < 2.40000000000000002e181

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 2.5%

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

   5. Taylor expanded in y around 0 27.3%

      \[\leadsto \color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 27.3%

         \[\leadsto 1 + \color{blue}{{y}^{2} \cdot -0.16666666666666666} \]

      2. unpow2 27.3%

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

   7. Simplified 27.3%

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

   if 2.40000000000000002e181 < x

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. associate-/l/ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. associate-/r* 100.0%

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

      3. associate-/l/ 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

   10. Simplified 40.7%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 800:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+181}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{-2 \cdot \left(x \cdot x\right)}{y}\\ \end{array} \]

Alternative 5: 36.1% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+181}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{-2 \cdot \left(x \cdot x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.2e+181)
   (+ 1.0 (* (* y y) -0.16666666666666666))
   (+ -2.0 (/ (* -2.0 (* x x)) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.2e+181) {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	} else {
		tmp = -2.0 + ((-2.0 * (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 <= 2.2d+181) then
        tmp = 1.0d0 + ((y * y) * (-0.16666666666666666d0))
    else
        tmp = (-2.0d0) + (((-2.0d0) * (x * x)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.2e+181) {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	} else {
		tmp = -2.0 + ((-2.0 * (x * x)) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.2e+181:
		tmp = 1.0 + ((y * y) * -0.16666666666666666)
	else:
		tmp = -2.0 + ((-2.0 * (x * x)) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.2e+181)
		tmp = Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = Float64(-2.0 + Float64(Float64(-2.0 * Float64(x * x)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.2e+181)
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	else
		tmp = -2.0 + ((-2.0 * (x * x)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.2e+181], N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(-2.0 + N[(N[(-2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000001e181

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 52.2%

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

   5. Taylor expanded in y around 0 37.1%

      \[\leadsto \color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 37.1%

         \[\leadsto 1 + \color{blue}{{y}^{2} \cdot -0.16666666666666666} \]

      2. unpow2 37.1%

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

   7. Simplified 37.1%

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

   if 2.2000000000000001e181 < x

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. associate-/l/ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. associate-/r* 100.0%

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

      3. associate-/l/ 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

   10. Simplified 40.7%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+181}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{-2 \cdot \left(x \cdot x\right)}{y}\\ \end{array} \]

Alternative 6: 32.5% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + \left(y \cdot y\right) \cdot -0.16666666666666666 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* (* y y) -0.16666666666666666)))

C

double code(double x, double y) {
	return 1.0 + ((y * y) * -0.16666666666666666);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((y * y) * (-0.16666666666666666d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((y * y) * -0.16666666666666666);
}

Python

def code(x, y):
	return 1.0 + ((y * y) * -0.16666666666666666)

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((y * y) * -0.16666666666666666);
end

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 47.0%

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

5. Taylor expanded in y around 0 36.1%

   \[\leadsto \color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}} \]
6. Step-by-step derivation

   1. *-commutative 36.1%

      \[\leadsto 1 + \color{blue}{{y}^{2} \cdot -0.16666666666666666} \]

   2. unpow2 36.1%

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

7. Simplified 36.1%

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

8. Final simplification 36.1%

   \[\leadsto 1 + \left(y \cdot y\right) \cdot -0.16666666666666666 \]

Alternative 7: 2.0% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -2.0)

C

double code(double x, double y) {
	return -2.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -2.0d0
end function

Java

public static double code(double x, double y) {
	return -2.0;
}

Python

def code(x, y):
	return -2.0

Julia

function code(x, y)
	return -2.0
end

MATLAB

function tmp = code(x, y)
	tmp = -2.0;
end

Wolfram

code[x_, y_] := -2.0

Derivation

1. Initial program 99.9%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

   3. associate-/l/ 99.9%

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

5. Applied egg-rr 99.9%

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

   1. unpow-1 99.9%

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

   2. associate-/r* 99.9%

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

   3. associate-/l/ 99.9%

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

7. Applied egg-rr 99.9%

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

8. Taylor expanded in x around 0 47.0%

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

10. Simplified 1.9%

    \[\leadsto \color{blue}{-2} \]

11. Final simplification 1.9%

    \[\leadsto -2 \]

Alternative 8: 7.4% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.8333333333333334 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.8333333333333334)

C

double code(double x, double y) {
	return 0.8333333333333334;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.8333333333333334d0
end function

Java

public static double code(double x, double y) {
	return 0.8333333333333334;
}

Python

def code(x, y):
	return 0.8333333333333334

Julia

function code(x, y)
	return 0.8333333333333334
end

MATLAB

function tmp = code(x, y)
	tmp = 0.8333333333333334;
end

Wolfram

code[x_, y_] := 0.8333333333333334

Derivation

1. Initial program 99.9%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 47.0%

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

   1. clear-num 46.9%

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

   2. inv-pow 46.9%

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

6. Applied egg-rr 46.9%

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

7. Taylor expanded in y around 0 36.1%

   \[\leadsto \color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}} \]

9. Simplified 7.6%

   \[\leadsto \color{blue}{0.8333333333333334} \]

10. Final simplification 7.6%

    \[\leadsto 0.8333333333333334 \]

Alternative 9: 26.4% 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. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 47.0%

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

   1. clear-num 46.9%

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

   2. inv-pow 46.9%

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

6. Applied egg-rr 46.9%

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

7. Taylor expanded in y around 0 27.1%

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

8. Final simplification 27.1%

   \[\leadsto 1 \]

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 2023297 
(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)))