Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 7.5s

Alternatives: 5

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.00005:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.00005) (/ (sin y) y) (* (cosh x) (/ y y))))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.00005) {
		tmp = sin(y) / y;
	} else {
		tmp = cosh(x) * (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 (cosh(x) <= 1.00005d0) then
        tmp = sin(y) / y
    else
        tmp = cosh(x) * (y / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.cosh(x) <= 1.00005) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x) * (y / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.00005:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x) * (y / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.00005)
		tmp = Float64(sin(y) / y);
	else
		tmp = Float64(cosh(x) * Float64(y / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (cosh(x) <= 1.00005)
		tmp = sin(y) / y;
	else
		tmp = cosh(x) * (y / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.00005], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(y / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.00005000000000011

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.6%

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

   if 1.00005000000000011 < (cosh.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 68.1%

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

Alternative 3: 53.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1250:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1250.0) (/ (sin y) y) (* -0.16666666666666666 (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1250.0) {
		tmp = sin(y) / y;
	} else {
		tmp = -0.16666666666666666 * (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 (x <= 1250.0d0) then
        tmp = sin(y) / y
    else
        tmp = (-0.16666666666666666d0) * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1250.0) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = -0.16666666666666666 * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1250.0:
		tmp = math.sin(y) / y
	else:
		tmp = -0.16666666666666666 * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1250.0)
		tmp = Float64(sin(y) / y);
	else
		tmp = Float64(-0.16666666666666666 * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1250.0)
		tmp = sin(y) / y;
	else
		tmp = -0.16666666666666666 * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1250.0], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1250

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 72.5%

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

   if 1250 < 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-*l/ 100.0%

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

      3. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 2.7%

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

   6. Taylor expanded in y around 0 19.9%

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

      1. distribute-rgt-in 19.9%

         \[\leadsto \color{blue}{\left(1 \cdot y + \left(-0.16666666666666666 \cdot {y}^{2}\right) \cdot y\right)} \cdot \frac{1}{y} \]

      2. *-lft-identity 19.9%

         \[\leadsto \left(\color{blue}{y} + \left(-0.16666666666666666 \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{1}{y} \]

      3. associate-*l* 19.9%

         \[\leadsto \left(y + \color{blue}{-0.16666666666666666 \cdot \left({y}^{2} \cdot y\right)}\right) \cdot \frac{1}{y} \]

      4. unpow2 19.9%

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

      5. unpow3 19.9%

         \[\leadsto \left(y + -0.16666666666666666 \cdot \color{blue}{{y}^{3}}\right) \cdot \frac{1}{y} \]

   8. Simplified 19.9%

      \[\leadsto \color{blue}{\left(y + -0.16666666666666666 \cdot {y}^{3}\right)} \cdot \frac{1}{y} \]

   9. Taylor expanded in y around inf 14.6%

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

       1. unpow2 14.6%

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

   11. Applied egg-rr 14.6%

       \[\leadsto -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 28.9% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1250:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1250.0) 1.0 (* -0.16666666666666666 (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1250.0) {
		tmp = 1.0;
	} else {
		tmp = -0.16666666666666666 * (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 (x <= 1250.0d0) then
        tmp = 1.0d0
    else
        tmp = (-0.16666666666666666d0) * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1250.0) {
		tmp = 1.0;
	} else {
		tmp = -0.16666666666666666 * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1250.0:
		tmp = 1.0
	else:
		tmp = -0.16666666666666666 * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1250.0)
		tmp = 1.0;
	else
		tmp = Float64(-0.16666666666666666 * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1250.0)
		tmp = 1.0;
	else
		tmp = -0.16666666666666666 * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1250.0], 1.0, N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1250

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 72.5%

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

   4. Taylor expanded in y around 0 39.4%

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

   5. Taylor expanded in y around 0 39.4%

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

   if 1250 < 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-*l/ 100.0%

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

      3. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 2.7%

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

   6. Taylor expanded in y around 0 19.9%

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

      1. distribute-rgt-in 19.9%

         \[\leadsto \color{blue}{\left(1 \cdot y + \left(-0.16666666666666666 \cdot {y}^{2}\right) \cdot y\right)} \cdot \frac{1}{y} \]

      2. *-lft-identity 19.9%

         \[\leadsto \left(\color{blue}{y} + \left(-0.16666666666666666 \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{1}{y} \]

      3. associate-*l* 19.9%

         \[\leadsto \left(y + \color{blue}{-0.16666666666666666 \cdot \left({y}^{2} \cdot y\right)}\right) \cdot \frac{1}{y} \]

      4. unpow2 19.9%

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

      5. unpow3 19.9%

         \[\leadsto \left(y + -0.16666666666666666 \cdot \color{blue}{{y}^{3}}\right) \cdot \frac{1}{y} \]

   8. Simplified 19.9%

      \[\leadsto \color{blue}{\left(y + -0.16666666666666666 \cdot {y}^{3}\right)} \cdot \frac{1}{y} \]

   9. Taylor expanded in y around inf 14.6%

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

       1. unpow2 14.6%

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

   11. Applied egg-rr 14.6%

       \[\leadsto -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 26.2% 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. Add Preprocessing

3. Taylor expanded in x around 0 55.8%

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

4. Taylor expanded in y around 0 30.6%

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

5. Taylor expanded in y around 0 30.6%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024123 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (* (cosh x) (sin y)) y))

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