Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 16.4s

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

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

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. Final simplification 99.9%

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

Alternative 2: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.002:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.002) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.002) {
		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) <= 1.002d0) 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) <= 1.002) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.002:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.002)
		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) <= 1.002)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.002], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.002

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.1%

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

   if 1.002 < (cosh.f64 x)

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{\frac{y}{\sin y}}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 99.9%

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

      2. div-inv 99.9%

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

      3. times-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0 81.2%

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

      1. frac-times 81.1%

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

      2. add-sqr-sqrt 81.2%

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

      3. rgt-mult-inverse 81.2%

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

   9. Applied egg-rr 81.2%

      \[\leadsto \color{blue}{\frac{\cosh x}{1}} \]
   10. Step-by-step derivation

       1. /-rgt-identity 81.2%

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

   11. Simplified 81.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.002:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.0% 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. Add Preprocessing
3. Step-by-step derivation

   1. clear-num 99.9%

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

   2. un-div-inv 99.9%

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

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\cosh x}{\frac{y}{\sin y}}} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 99.9%

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

   2. div-inv 99.8%

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

   3. times-frac 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in y around 0 70.2%

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

   1. frac-times 70.1%

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

   2. add-sqr-sqrt 70.1%

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

   3. rgt-mult-inverse 70.2%

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

9. Applied egg-rr 70.2%

   \[\leadsto \color{blue}{\frac{\cosh x}{1}} \]
10. Step-by-step derivation

    1. /-rgt-identity 70.2%

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

11. Simplified 70.2%

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

12. Final simplification 70.2%

    \[\leadsto \cosh x \]
13. Add Preprocessing

Alternative 4: 45.4% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + 0.5 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* 0.5 (* x x))))

C

double code(double x, double y) {
	return 1.0 + (0.5 * (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (0.5d0 * (x * x))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (0.5 * (x * x));
}

Python

def code(x, y):
	return 1.0 + (0.5 * (x * x))

Julia

function code(x, y)
	return Float64(1.0 + Float64(0.5 * Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (0.5 * (x * x));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0 70.2%

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

4. Taylor expanded in x around 0 49.9%

   \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {x}^{2}\right)} \cdot 1 \]
5. Step-by-step derivation

   1. unpow2 49.9%

      \[\leadsto \left(1 + 0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot 1 \]

6. Applied egg-rr 49.9%

   \[\leadsto \left(1 + 0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot 1 \]

7. Final simplification 49.9%

   \[\leadsto 1 + 0.5 \cdot \left(x \cdot x\right) \]
8. 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. 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 51.5%

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

6. Taylor expanded in y around 0 34.8%

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

   1. *-commutative 34.8%

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

8. Simplified 34.8%

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

9. Taylor expanded in y around 0 31.4%

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

10. Final simplification 31.4%

    \[\leadsto 1 \]
11. 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 2024076 
(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)))