Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.8% → 99.9%

Time: 1.1s

Alternatives: 4

Speedup: 0.9×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ (* (sin x) (sinh y)) x))

C

double code(double x, double y) {
	return (sin(x) * sinh(y)) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (sin(x) * sinh(y)) / x
end function

Java

public static double code(double x, double y) {
	return (Math.sin(x) * Math.sinh(y)) / x;
}

Python

def code(x, y):
	return (math.sin(x) * math.sinh(y)) / x

Julia

function code(x, y)
	return Float64(Float64(sin(x) * sinh(y)) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = (sin(x) * sinh(y)) / x;
end

Wolfram

code[x_, y_] := N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Initial Program: 88.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ (* (sin x) (sinh y)) x))

C

double code(double x, double y) {
	return (sin(x) * sinh(y)) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (sin(x) * sinh(y)) / x
end function

Java

public static double code(double x, double y) {
	return (Math.sin(x) * Math.sinh(y)) / x;
}

Python

def code(x, y):
	return (math.sin(x) * math.sinh(y)) / x

Julia

function code(x, y)
	return Float64(Float64(sin(x) * sinh(y)) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = (sin(x) * sinh(y)) / x;
end

Wolfram

code[x_, y_] := N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10^{-25}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= x -1.5e-8) t_0 (if (<= x 1e-25) (sinh y) t_0))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (x <= -1.5e-8) {
		tmp = t_0;
	} else if (x <= 1e-25) {
		tmp = sinh(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(x) * sinh(y)) / x
    if (x <= (-1.5d-8)) then
        tmp = t_0
    else if (x <= 1d-25) then
        tmp = sinh(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (Math.sin(x) * Math.sinh(y)) / x;
	double tmp;
	if (x <= -1.5e-8) {
		tmp = t_0;
	} else if (x <= 1e-25) {
		tmp = Math.sinh(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sin(x) * math.sinh(y)) / x
	tmp = 0
	if x <= -1.5e-8:
		tmp = t_0
	elif x <= 1e-25:
		tmp = math.sinh(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (x <= -1.5e-8)
		tmp = t_0;
	elseif (x <= 1e-25)
		tmp = sinh(y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sin(x) * sinh(y)) / x;
	tmp = 0.0;
	if (x <= -1.5e-8)
		tmp = t_0;
	elseif (x <= 1e-25)
		tmp = sinh(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -1.5e-8], t$95$0, If[LessEqual[x, 1e-25], N[Sinh[y], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999987e-8 or 1.00000000000000004e-25 < x

   1. Initial program 99.9%

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

   if -1.49999999999999987e-8 < x < 1.00000000000000004e-25

   1. Initial program 75.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + {x}^{2} \cdot \left(\frac{-1}{12} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + {x}^{2} \cdot \left(\frac{-1}{10080} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{240} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right)\right)} \]

   5. Applied rewrites 100.0%

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

Alternative 2: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.88) (sinh y) (* (sin x) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.88) {
		tmp = sinh(y);
	} else {
		tmp = sin(x) * sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.88d0) then
        tmp = sinh(y)
    else
        tmp = sin(x) * sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.88) {
		tmp = Math.sinh(y);
	} else {
		tmp = Math.sin(x) * Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.88:
		tmp = math.sinh(y)
	else:
		tmp = math.sin(x) * math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.88)
		tmp = sinh(y);
	else
		tmp = Float64(sin(x) * sinh(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.88)
		tmp = sinh(y);
	else
		tmp = sin(x) * sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.88], N[Sinh[y], $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + {x}^{2} \cdot \left(\frac{-1}{12} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + {x}^{2} \cdot \left(\frac{-1}{10080} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{240} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right)\right)} \]

   5. Applied rewrites 77.1%

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

   if 0.880000000000000004 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]

   5. Applied rewrites 59.4%

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

Alternative 3: 63.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sinh y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sinh y))

C

double code(double x, double y) {
	return sinh(y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sinh(y)
end function

Java

public static double code(double x, double y) {
	return Math.sinh(y);
}

Python

def code(x, y):
	return math.sinh(y)

Julia

function code(x, y)
	return sinh(y)
end

MATLAB

function tmp = code(x, y)
	tmp = sinh(y);
end

Wolfram

code[x_, y_] := N[Sinh[y], $MachinePrecision]

Derivation

1. Initial program 88.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + {x}^{2} \cdot \left(\frac{-1}{12} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + {x}^{2} \cdot \left(\frac{-1}{10080} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{240} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right)\right)} \]

5. Applied rewrites 67.6%

   \[\leadsto \color{blue}{\sinh y} \]
6. Add Preprocessing

Alternative 4: 2.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sin x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sin x))

C

double code(double x, double y) {
	return sin(x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x);
}

Python

def code(x, y):
	return math.sin(x)

Julia

function code(x, y)
	return sin(x)
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x);
end

Wolfram

code[x_, y_] := N[Sin[x], $MachinePrecision]

Derivation

1. Initial program 88.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + {x}^{2} \cdot \left(\frac{-1}{12} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \frac{1}{240} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right)\right)} \]

5. Applied rewrites 3.0%

   \[\leadsto \color{blue}{\sin x} \]
6. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) x)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / x)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / x);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / x)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / x);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024321 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64
  :pre (TRUE)

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

  (/ (* (sin x) (sinh y)) x))