Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 8.5s

Alternatives: 5

Speedup: 1.0×

• 48.5% 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 99.8%

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

Alternative 2: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.00000000002:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.00000000002) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.00000000002) {
		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) <= 1.00000000002d0) 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) <= 1.00000000002) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.00000000002:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.00000000002)
		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) <= 1.00000000002)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.00000000002], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.00000000002

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.6%

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

   if 1.00000000002 < (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 74.7%

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

      1. *-inverses 74.7%

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

      2. *-rgt-identity 74.7%

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

      3. cosh-def 74.7%

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

      4. clear-num 74.7%

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

      5. cosh-undef 74.7%

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

   5. Applied egg-rr 74.7%

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

      1. associate-/r* 74.7%

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

      2. metadata-eval 74.7%

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

      3. remove-double-div 74.7%

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

   7. Simplified 74.7%

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

Alternative 3: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+234}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.4e+234) (cosh x) (* -0.16666666666666666 (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.4e+234) {
		tmp = cosh(x);
	} else {
		tmp = -0.16666666666666666 * (y * 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 <= 6.4d+234) then
        tmp = cosh(x)
    else
        tmp = (-0.16666666666666666d0) * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.4e+234) {
		tmp = Math.cosh(x);
	} else {
		tmp = -0.16666666666666666 * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.4e+234:
		tmp = math.cosh(x)
	else:
		tmp = -0.16666666666666666 * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.4e+234)
		tmp = cosh(x);
	else
		tmp = Float64(-0.16666666666666666 * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.4e+234)
		tmp = cosh(x);
	else
		tmp = -0.16666666666666666 * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.4e+234], N[Cosh[x], $MachinePrecision], N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.39999999999999983e234

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 66.6%

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

      1. *-inverses 66.6%

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

      2. *-rgt-identity 66.6%

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

      3. cosh-def 66.6%

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

      4. clear-num 66.6%

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

      5. cosh-undef 66.6%

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

   5. Applied egg-rr 66.6%

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

      1. associate-/r* 66.6%

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

      2. metadata-eval 66.6%

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

      3. remove-double-div 66.6%

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

   7. Simplified 66.6%

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

   if 6.39999999999999983e234 < y

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 31.3%

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

      1. distribute-rgt-in 31.3%

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

      2. *-lft-identity 31.3%

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

      3. associate-*l* 31.3%

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

      4. unpow2 31.3%

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

      5. unpow3 31.3%

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

   6. Simplified 31.3%

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

   7. Taylor expanded in y around inf 31.3%

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

      1. unpow2 31.3%

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

   9. Applied egg-rr 31.3%

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

Alternative 4: 29.9% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{+142}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.3e+142) 1.0 (* -0.16666666666666666 (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.3e+142) {
		tmp = 1.0;
	} else {
		tmp = -0.16666666666666666 * (y * 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 <= 4.3d+142) then
        tmp = 1.0d0
    else
        tmp = (-0.16666666666666666d0) * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.3e+142) {
		tmp = 1.0;
	} else {
		tmp = -0.16666666666666666 * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.3e+142:
		tmp = 1.0
	else:
		tmp = -0.16666666666666666 * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.3e+142)
		tmp = 1.0;
	else
		tmp = Float64(-0.16666666666666666 * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.3e+142)
		tmp = 1.0;
	else
		tmp = -0.16666666666666666 * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.3e+142], 1.0, N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.30000000000000012e142

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 68.6%

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

   4. Taylor expanded in x around 0 29.8%

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

   if 4.30000000000000012e142 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 47.6%

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

   4. Taylor expanded in y around 0 31.3%

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

      1. distribute-rgt-in 31.3%

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

      2. *-lft-identity 31.3%

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

      3. associate-*l* 31.3%

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

      4. unpow2 31.3%

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

      5. unpow3 31.3%

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

   6. Simplified 31.3%

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

   7. Taylor expanded in y around inf 26.4%

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

      1. unpow2 26.4%

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

   9. Applied egg-rr 26.4%

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

Alternative 5: 27.4% 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 99.8%

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

3. Taylor expanded in y around 0 62.3%

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

4. Taylor expanded in x around 0 25.9%

   \[\leadsto \color{blue}{1} \]
5. 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 2024172 
(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)))