Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 8.5s

Alternatives: 6

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. Add Preprocessing

Alternative 2: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 195000:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 195000.0) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 195000.0) {
		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 (cosh(x) <= 195000.0d0) 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 (Math.cosh(x) <= 195000.0) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 195000.0:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 195000.0)
		tmp = Float64(sin(y) / y);
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (cosh(x) <= 195000.0)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 195000.0], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 195000

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 97.9%

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

   if 195000 < (cosh.f64 x)

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \cosh x \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin y\right)\right)}}{y} \]

      2. expm1-undefine 61.8%

         \[\leadsto \cosh x \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\sin y\right)} - 1}}{y} \]

   4. Applied egg-rr 61.8%

      \[\leadsto \cosh x \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\sin y\right)} - 1}}{y} \]
   5. Step-by-step derivation

      1. expm1-define 100.0%

         \[\leadsto \cosh x \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin y\right)\right)}}{y} \]

   6. Simplified 100.0%

      \[\leadsto \cosh x \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin y\right)\right)}}{y} \]

   7. Taylor expanded in y around 0 73.2%

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

      1. *-rgt-identity 73.2%

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

      2. cosh-def 73.2%

         \[\leadsto \color{blue}{\frac{e^{x} + e^{-x}}{2}} \]

      3. clear-num 73.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{2}{e^{x} + e^{-x}}}} \]

      4. cosh-undef 73.2%

         \[\leadsto \frac{1}{\frac{2}{\color{blue}{2 \cdot \cosh x}}} \]

   9. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{2 \cdot \cosh x}}} \]
   10. Step-by-step derivation

       1. associate-/r* 73.2%

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{2}{2}}{\cosh x}}} \]

       2. metadata-eval 73.2%

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

       3. remove-double-div 73.2%

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

   11. Simplified 73.2%

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

Alternative 3: 64.0% 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. expm1-log1p-u 99.9%

      \[\leadsto \cosh x \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin y\right)\right)}}{y} \]

   2. expm1-undefine 54.1%

      \[\leadsto \cosh x \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\sin y\right)} - 1}}{y} \]

4. Applied egg-rr 54.1%

   \[\leadsto \cosh x \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\sin y\right)} - 1}}{y} \]
5. Step-by-step derivation

   1. expm1-define 99.9%

      \[\leadsto \cosh x \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin y\right)\right)}}{y} \]

6. Simplified 99.9%

   \[\leadsto \cosh x \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin y\right)\right)}}{y} \]

7. Taylor expanded in y around 0 64.6%

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

   1. *-rgt-identity 64.6%

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

   2. cosh-def 64.6%

      \[\leadsto \color{blue}{\frac{e^{x} + e^{-x}}{2}} \]

   3. clear-num 64.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{e^{x} + e^{-x}}}} \]

   4. cosh-undef 64.6%

      \[\leadsto \frac{1}{\frac{2}{\color{blue}{2 \cdot \cosh x}}} \]

9. Applied egg-rr 64.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{2}{2 \cdot \cosh x}}} \]
10. Step-by-step derivation

    1. associate-/r* 64.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{2}{2}}{\cosh x}}} \]

    2. metadata-eval 64.6%

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

    3. remove-double-div 64.6%

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

11. Simplified 64.6%

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

Alternative 4: 30.2% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+113}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.2e+113) 1.0 (* y (* y -0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.2e+113) {
		tmp = 1.0;
	} else {
		tmp = 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 <= 1.2d+113) then
        tmp = 1.0d0
    else
        tmp = y * (y * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.2e+113) {
		tmp = 1.0;
	} else {
		tmp = y * (y * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.2e+113:
		tmp = 1.0
	else:
		tmp = y * (y * -0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.2e+113)
		tmp = 1.0;
	else
		tmp = Float64(y * Float64(y * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.2e+113)
		tmp = 1.0;
	else
		tmp = y * (y * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.2e+113], 1.0, N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.19999999999999992e113

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 53.6%

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

   4. Taylor expanded in y around 0 34.0%

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

   5. Taylor expanded in y around 0 34.0%

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

   if 1.19999999999999992e113 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 41.0%

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

   4. Taylor expanded in y around 0 37.3%

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

      1. distribute-rgt-in 37.3%

         \[\leadsto \frac{\color{blue}{1 \cdot y + \left(-0.16666666666666666 \cdot {y}^{2}\right) \cdot y}}{y} \]

      2. *-lft-identity 37.3%

         \[\leadsto \frac{\color{blue}{y} + \left(-0.16666666666666666 \cdot {y}^{2}\right) \cdot y}{y} \]

      3. associate-*l* 37.3%

         \[\leadsto \frac{y + \color{blue}{-0.16666666666666666 \cdot \left({y}^{2} \cdot y\right)}}{y} \]

      4. unpow2 37.3%

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

      5. unpow3 37.3%

         \[\leadsto \frac{y + -0.16666666666666666 \cdot \color{blue}{{y}^{3}}}{y} \]

   6. Simplified 37.3%

      \[\leadsto \frac{\color{blue}{y + -0.16666666666666666 \cdot {y}^{3}}}{y} \]

   7. Taylor expanded in y around inf 37.3%

      \[\leadsto \frac{\color{blue}{-0.16666666666666666 \cdot {y}^{3}}}{y} \]
   8. Step-by-step derivation

      1. associate-/l* 37.3%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{{y}^{3}}{y}} \]

      2. pow1 37.3%

         \[\leadsto -0.16666666666666666 \cdot \frac{{y}^{3}}{\color{blue}{{y}^{1}}} \]

      3. pow-div 28.0%

         \[\leadsto -0.16666666666666666 \cdot \color{blue}{{y}^{\left(3 - 1\right)}} \]

      4. metadata-eval 28.0%

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

      5. pow2 28.0%

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

      6. associate-*r* 28.0%

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

      7. *-commutative 28.0%

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

   9. Applied egg-rr 28.0%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+113}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.9% 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. Taylor expanded in x around 0 52.1%

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

4. Taylor expanded in y around 0 34.9%

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

   1. *-commutative 34.9%

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

6. Simplified 34.9%

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

   1. unpow2 34.9%

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

8. Applied egg-rr 34.9%

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

Alternative 6: 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. Add Preprocessing

3. Taylor expanded in x around 0 52.1%

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

4. Taylor expanded in y around 0 30.3%

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

5. Taylor expanded in y around 0 30.3%

   \[\leadsto \color{blue}{1} \]
6. 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 2024133 
(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)))