Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 8.2s

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

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.00005) {
		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 (cosh(x) <= 1.00005d0) 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 (Math.cosh(x) <= 1.00005) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	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)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.00005)
		tmp = Float64(sin(y) / y);
	else
		tmp = cosh(x);
	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);
	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[Cosh[x], $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%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.00005:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{+158}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{y \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.8e+158)
   (cosh x)
   (/ 1.0 (/ y (* y (+ 1.0 (* -0.16666666666666666 (* y y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.8e+158) {
		tmp = cosh(x);
	} else {
		tmp = 1.0 / (y / (y * (1.0 + (-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 (y <= 7.8d+158) then
        tmp = cosh(x)
    else
        tmp = 1.0d0 / (y / (y * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.8e+158) {
		tmp = Math.cosh(x);
	} else {
		tmp = 1.0 / (y / (y * (1.0 + (-0.16666666666666666 * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.8e+158:
		tmp = math.cosh(x)
	else:
		tmp = 1.0 / (y / (y * (1.0 + (-0.16666666666666666 * (y * y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.8e+158)
		tmp = cosh(x);
	else
		tmp = Float64(1.0 / Float64(y / Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.8e+158)
		tmp = cosh(x);
	else
		tmp = 1.0 / (y / (y * (1.0 + (-0.16666666666666666 * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.8e+158], N[Cosh[x], $MachinePrecision], N[(1.0 / N[(y / N[(y * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.8e158

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 67.0%

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

   if 7.8e158 < y

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 44.9%

      \[\leadsto \sin y \cdot \color{blue}{\frac{1}{y}} \]
   6. Step-by-step derivation

      1. div-inv 45.0%

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

      2. clear-num 44.9%

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

   7. Applied egg-rr 44.9%

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

   8. Taylor expanded in y around 0 41.9%

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

      1. unpow2 41.9%

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

   10. Applied egg-rr 41.9%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{+158}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{y \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 33.9% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-*l/ 99.9%

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

   3. associate-/l* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 55.7%

   \[\leadsto \sin y \cdot \color{blue}{\frac{1}{y}} \]
6. Step-by-step derivation

   1. div-inv 55.8%

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

   2. clear-num 55.8%

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

7. Applied egg-rr 55.8%

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

8. Taylor expanded in y around 0 39.4%

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

   1. unpow2 39.4%

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

10. Applied egg-rr 39.4%

    \[\leadsto \frac{1}{\frac{y}{y \cdot \left(1 + -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right)}} \]
11. 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%

   \[\leadsto \color{blue}{1} \]
5. 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)))