Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 7.8s

Alternatives: 5

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

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

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: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.000002:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{\frac{1}{y}}{\cosh x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.000002)
   (/ (sin y) y)
   (/ 1.0 (* y (/ (/ 1.0 y) (cosh x))))))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.000002) {
		tmp = sin(y) / y;
	} else {
		tmp = 1.0 / (y * ((1.0 / y) / 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) <= 1.000002d0) then
        tmp = sin(y) / y
    else
        tmp = 1.0d0 / (y * ((1.0d0 / y) / cosh(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.cosh(x) <= 1.000002) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = 1.0 / (y * ((1.0 / y) / Math.cosh(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.000002:
		tmp = math.sin(y) / y
	else:
		tmp = 1.0 / (y * ((1.0 / y) / math.cosh(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.000002)
		tmp = Float64(sin(y) / y);
	else
		tmp = Float64(1.0 / Float64(y * Float64(Float64(1.0 / y) / cosh(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (cosh(x) <= 1.000002)
		tmp = sin(y) / y;
	else
		tmp = 1.0 / (y * ((1.0 / y) / cosh(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.000002], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(y * N[(N[(1.0 / y), $MachinePrecision] / N[Cosh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.00000200000000006

   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* 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

   if 1.00000200000000006 < (cosh.f64 x)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

      2. clear-num 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. associate-/r* 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in y around 0 70.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.000002:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{\frac{1}{y}}{\cosh x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.000002:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.000002) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.000002) {
		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) <= 1.000002d0) 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) <= 1.000002) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.000002:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.000002)
		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) <= 1.000002)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.000002], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.00000200000000006

   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* 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

   if 1.00000200000000006 < (cosh.f64 x)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

      2. clear-num 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. associate-/r* 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in y around 0 70.6%

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

       1. inv-pow 70.6%

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

       2. metadata-eval 70.6%

          \[\leadsto \frac{1}{\frac{{y}^{\color{blue}{\left(\frac{-2}{2}\right)}}}{\cosh x} \cdot y} \]

       3. sqrt-pow2 29.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{{\left(\sqrt{y}\right)}^{-2}}}{\cosh x} \cdot y} \]

       4. associate-*l/ 29.4%

          \[\leadsto \frac{1}{\color{blue}{\frac{{\left(\sqrt{y}\right)}^{-2} \cdot y}{\cosh x}}} \]

       5. sqrt-pow2 70.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{{y}^{\left(\frac{-2}{2}\right)}} \cdot y}{\cosh x}} \]

       6. metadata-eval 70.6%

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

       7. inv-pow 70.6%

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

       8. lft-mult-inverse 70.6%

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

       9. associate-/r/ 70.6%

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

       10. metadata-eval 70.6%

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

   11. Applied egg-rr 70.6%

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

       1. *-lft-identity 70.6%

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

   13. Simplified 70.6%

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

Alternative 4: 63.6% 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. 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. Step-by-step derivation

   1. associate-*r/ 99.9%

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

   2. clear-num 99.5%

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

6. Applied egg-rr 99.5%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.4%

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

   3. associate-/r* 99.4%

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

8. Applied egg-rr 99.4%

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

9. Taylor expanded in y around 0 59.6%

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

    1. inv-pow 59.6%

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

    2. metadata-eval 59.6%

       \[\leadsto \frac{1}{\frac{{y}^{\color{blue}{\left(\frac{-2}{2}\right)}}}{\cosh x} \cdot y} \]

    3. sqrt-pow2 28.1%

       \[\leadsto \frac{1}{\frac{\color{blue}{{\left(\sqrt{y}\right)}^{-2}}}{\cosh x} \cdot y} \]

    4. associate-*l/ 28.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{{\left(\sqrt{y}\right)}^{-2} \cdot y}{\cosh x}}} \]

    5. sqrt-pow2 59.6%

       \[\leadsto \frac{1}{\frac{\color{blue}{{y}^{\left(\frac{-2}{2}\right)}} \cdot y}{\cosh x}} \]

    6. metadata-eval 59.6%

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

    7. inv-pow 59.6%

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

    8. lft-mult-inverse 59.7%

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

    9. associate-/r/ 59.7%

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

    10. metadata-eval 59.7%

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

11. Applied egg-rr 59.7%

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

    1. *-lft-identity 59.7%

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

13. Simplified 59.7%

    \[\leadsto \color{blue}{\cosh x} \]
14. Add Preprocessing

Alternative 5: 27.3% 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 54.7%

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

6. Taylor expanded in y around 0 28.2%

   \[\leadsto \color{blue}{1} \]
7. 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 2024101 
(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)))