Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 8.7s

Alternatives: 5

Speedup: 1.0×

• 49.5% 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. Add Preprocessing

3. Final simplification 99.9%

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

Alternative 2: 68.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 450000000:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 450000000.0) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 450000000.0) {
		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 (x <= 450000000.0d0) 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 (x <= 450000000.0) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 450000000.0:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 450000000.0)
		tmp = Float64(sin(y) / y);
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 450000000.0)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 450000000.0], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5e8

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 67.4%

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

   if 4.5e8 < 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 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 450000000:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+143}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9.2e+143) (cosh x) (+ 1.0 (* (* y y) -0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.2e+143) {
		tmp = cosh(x);
	} else {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.2d+143) then
        tmp = cosh(x)
    else
        tmp = 1.0d0 + ((y * y) * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9.2e+143) {
		tmp = Math.cosh(x);
	} else {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.2e+143:
		tmp = math.cosh(x)
	else:
		tmp = 1.0 + ((y * y) * -0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.2e+143)
		tmp = cosh(x);
	else
		tmp = Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.2e+143)
		tmp = cosh(x);
	else
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.2e+143], N[Cosh[x], $MachinePrecision], N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.1999999999999999e143

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 66.1%

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

   if 9.1999999999999999e143 < y

   1. Initial program 100.0%

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

      1. add-cube-cbrt 99.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\cosh x \cdot \frac{\sin y}{y}} \cdot \sqrt[3]{\cosh x \cdot \frac{\sin y}{y}}\right) \cdot \sqrt[3]{\cosh x \cdot \frac{\sin y}{y}}} \]

      2. pow3 99.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\cosh x \cdot \frac{\sin y}{y}}\right)}^{3}} \]

      3. *-commutative 99.1%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\sin y}{y} \cdot \cosh x}}\right)}^{3} \]

   4. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\sin y}{y} \cdot \cosh x}\right)}^{3}} \]

   5. Taylor expanded in x around 0 25.6%

      \[\leadsto {\color{blue}{\left({\left(\frac{\sin y}{y}\right)}^{0.3333333333333333}\right)}}^{3} \]
   6. Step-by-step derivation

      1. unpow1/3 44.1%

         \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{\sin y}{y}}\right)}}^{3} \]

   7. Simplified 44.1%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{\sin y}{y}}\right)}}^{3} \]

   8. Taylor expanded in y around 0 43.0%

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

      1. *-commutative 43.0%

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

   10. Simplified 43.0%

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

       1. unpow2 43.0%

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

   12. Applied egg-rr 43.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+143}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 32.4% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $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-cube-cbrt 99.4%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\cosh x \cdot \frac{\sin y}{y}} \cdot \sqrt[3]{\cosh x \cdot \frac{\sin y}{y}}\right) \cdot \sqrt[3]{\cosh x \cdot \frac{\sin y}{y}}} \]

   2. pow3 99.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cosh x \cdot \frac{\sin y}{y}}\right)}^{3}} \]

   3. *-commutative 99.4%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\sin y}{y} \cdot \cosh x}}\right)}^{3} \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\sin y}{y} \cdot \cosh x}\right)}^{3}} \]

5. Taylor expanded in x around 0 35.1%

   \[\leadsto {\color{blue}{\left({\left(\frac{\sin y}{y}\right)}^{0.3333333333333333}\right)}}^{3} \]
6. Step-by-step derivation

   1. unpow1/3 49.5%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{\sin y}{y}}\right)}}^{3} \]

7. Simplified 49.5%

   \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{\sin y}{y}}\right)}}^{3} \]

8. Taylor expanded in y around 0 33.1%

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

   1. *-commutative 33.1%

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

10. Simplified 33.1%

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

    1. unpow2 33.1%

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

12. Applied egg-rr 33.1%

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

13. Final simplification 33.1%

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

Alternative 5: 26.5% 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. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 99.4%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\cosh x \cdot \frac{\sin y}{y}} \cdot \sqrt[3]{\cosh x \cdot \frac{\sin y}{y}}\right) \cdot \sqrt[3]{\cosh x \cdot \frac{\sin y}{y}}} \]

   2. pow3 99.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cosh x \cdot \frac{\sin y}{y}}\right)}^{3}} \]

   3. *-commutative 99.4%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\sin y}{y} \cdot \cosh x}}\right)}^{3} \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\sin y}{y} \cdot \cosh x}\right)}^{3}} \]

5. Taylor expanded in x around 0 35.1%

   \[\leadsto {\color{blue}{\left({\left(\frac{\sin y}{y}\right)}^{0.3333333333333333}\right)}}^{3} \]
6. Step-by-step derivation

   1. unpow1/3 49.5%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{\sin y}{y}}\right)}}^{3} \]

7. Simplified 49.5%

   \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{\sin y}{y}}\right)}}^{3} \]

8. Taylor expanded in y around 0 25.6%

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

9. Final simplification 25.6%

   \[\leadsto 1 \]
10. 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 2024020 
(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)))