Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 6.4s

Alternatives: 4

Speedup: 1.0×

• 50.0% 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. Final simplification 99.9%

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

Alternative 2: 63.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+63} \lor \neg \left(y \leq 1.8 \cdot 10^{+256}\right):\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 3e+63) (not (<= y 1.8e+256)))
   (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))
   (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 3e+63) || !(y <= 1.8e+256)) {
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (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 ((y <= 3d+63) .or. (.not. (y <= 1.8d+256))) then
        tmp = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    else
        tmp = cosh(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 3e+63) || !(y <= 1.8e+256)) {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 3e+63) or not (y <= 1.8e+256):
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 3e+63) || !(y <= 1.8e+256))
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 3e+63) || ~((y <= 1.8e+256)))
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 3e+63], N[Not[LessEqual[y, 1.8e+256]], $MachinePrecision]], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.99999999999999999e63 or 1.79999999999999985e256 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 70.1%

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

      1. unpow2 70.1%

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

   4. Simplified 70.1%

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

   if 2.99999999999999999e63 < y < 1.79999999999999985e256

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 43.6%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+63} \lor \neg \left(y \leq 1.8 \cdot 10^{+256}\right):\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 3: 63.7% 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 62.5%

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

3. Final simplification 62.5%

   \[\leadsto \cosh x \]

Alternative 4: 27.0% accurate, 22.8× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 1.0 / (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 / (1.0d0 + (y * (y * 0.16666666666666666d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-/r/ 99.5%

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

   2. div-inv 99.4%

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

   3. associate-/r* 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in y around 0 56.2%

   \[\leadsto \cosh x \cdot \frac{\frac{1}{y}}{\color{blue}{\frac{1}{y} + 0.16666666666666666 \cdot y}} \]
7. Step-by-step derivation

   1. *-commutative 56.2%

      \[\leadsto \cosh x \cdot \frac{\frac{1}{y}}{\frac{1}{y} + \color{blue}{y \cdot 0.16666666666666666}} \]

8. Simplified 56.2%

   \[\leadsto \cosh x \cdot \frac{\frac{1}{y}}{\color{blue}{\frac{1}{y} + y \cdot 0.16666666666666666}} \]

9. Taylor expanded in x around 0 28.0%

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

    1. *-commutative 28.0%

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

    2. *-commutative 28.0%

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

    3. distribute-rgt-in 28.0%

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

    4. lft-mult-inverse 28.1%

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

11. Simplified 28.1%

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

12. Final simplification 28.1%

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

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 2023258 
(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)))