Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.8%

Time: 11.8s

Alternatives: 7

Speedup: 1.0×

• 48.4% 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: 99.8% accurate, 1.0× speedup

Math

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

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(sin(y) / Float64(y / cosh(x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. add-sqr-sqrt 99.9%

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

   2. pow2 99.9%

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

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\sqrt{\cosh x}\right)}^{2}} \cdot \frac{\sin y}{y} \]
5. Step-by-step derivation

   1. unpow2 99.9%

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

   2. add-sqr-sqrt 99.9%

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

   3. associate-*r/ 99.9%

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

   4. associate-*l/ 99.8%

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

   5. clear-num 99.8%

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

   6. associate-*l/ 99.9%

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

   7. *-un-lft-identity 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.1:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.1) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.1) {
		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) <= 1.1d0) 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) <= 1.1) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.1:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.1)
		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) <= 1.1)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.1], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.1000000000000001

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 97.1%

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

   if 1.1000000000000001 < (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 65.3%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.1:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]
5. Add Preprocessing

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. Add Preprocessing
3. Add Preprocessing

Alternative 4: 63.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.1 \cdot 10^{+106}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.1e+106)
   (cosh x)
   (* (* y (+ 1.0 (* -0.16666666666666666 (* y y)))) (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.1e+106) {
		tmp = cosh(x);
	} else {
		tmp = (y * (1.0 + (-0.16666666666666666 * (y * y)))) * (1.0 / 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 <= 6.1d+106) then
        tmp = cosh(x)
    else
        tmp = (y * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.1e+106) {
		tmp = Math.cosh(x);
	} else {
		tmp = (y * (1.0 + (-0.16666666666666666 * (y * y)))) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.1e+106:
		tmp = math.cosh(x)
	else:
		tmp = (y * (1.0 + (-0.16666666666666666 * (y * y)))) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.1e+106)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)))) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.1e+106)
		tmp = cosh(x);
	else
		tmp = (y * (1.0 + (-0.16666666666666666 * (y * y)))) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.1e+106], N[Cosh[x], $MachinePrecision], N[(N[(y * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.10000000000000001e106

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 61.8%

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

   if 6.10000000000000001e106 < y

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      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 52.3%

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

   6. Taylor expanded in y around 0 24.6%

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

      1. unpow2 24.6%

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

   8. Applied egg-rr 24.6%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.1 \cdot 10^{+106}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.6% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $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 51.3%

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

6. Taylor expanded in y around 0 36.1%

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

   1. unpow2 36.1%

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

8. Applied egg-rr 36.1%

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

Alternative 6: 33.0% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $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 51.3%

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

6. Taylor expanded in y around 0 33.4%

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

   1. *-commutative 33.4%

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

8. Simplified 33.4%

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

   1. unpow2 36.1%

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

10. Applied egg-rr 33.4%

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

11. Final simplification 33.4%

    \[\leadsto 1 + -0.16666666666666666 \cdot \left(y \cdot y\right) \]
12. Add Preprocessing

Alternative 7: 27.5% 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. *-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 51.3%

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

6. Taylor expanded in y around 0 25.6%

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

7. Taylor expanded in y around 0 25.7%

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

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 2024110 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

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

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