Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 5.4s

Alternatives: 3

Speedup: 1.0×

• 49.0% 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

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

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 2: 62.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -750:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+148}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -750.0)
   (cosh x)
   (if (<= y 7.2e+148)
     (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))
     (cosh x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -750.0) {
		tmp = cosh(x);
	} else if (y <= 7.2e+148) {
		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 <= (-750.0d0)) then
        tmp = cosh(x)
    else if (y <= 7.2d+148) 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 <= -750.0) {
		tmp = Math.cosh(x);
	} else if (y <= 7.2e+148) {
		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 <= -750.0:
		tmp = math.cosh(x)
	elif y <= 7.2e+148:
		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 <= -750.0)
		tmp = cosh(x);
	elseif (y <= 7.2e+148)
		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 <= -750.0)
		tmp = cosh(x);
	elseif (y <= 7.2e+148)
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -750.0], N[Cosh[x], $MachinePrecision], If[LessEqual[y, 7.2e+148], 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 < -750 or 7.20000000000000013e148 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 31.0%

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

   if -750 < y < 7.20000000000000013e148

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 88.4%

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

      1. unpow2 88.4%

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

   4. Simplified 88.4%

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -750:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+148}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 3: 62.3% 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 63.9%

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

3. Final simplification 63.9%

   \[\leadsto \cosh x \]

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 2023182 
(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)))