Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 4

Speedup: 1.0×

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 54.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 780000000:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;1 + {x}^{2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 780000000.0) (cos x) (+ 1.0 (* (pow x 2.0) -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 780000000.0) {
		tmp = cos(x);
	} else {
		tmp = 1.0 + (pow(x, 2.0) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 780000000.0d0) then
        tmp = cos(x)
    else
        tmp = 1.0d0 + ((x ** 2.0d0) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 780000000.0) {
		tmp = Math.cos(x);
	} else {
		tmp = 1.0 + (Math.pow(x, 2.0) * -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 780000000.0:
		tmp = math.cos(x)
	else:
		tmp = 1.0 + (math.pow(x, 2.0) * -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 780000000.0)
		tmp = cos(x);
	else
		tmp = Float64(1.0 + Float64((x ^ 2.0) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 780000000.0)
		tmp = cos(x);
	else
		tmp = 1.0 + ((x ^ 2.0) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 780000000.0], N[Cos[x], $MachinePrecision], N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.8e8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 61.5%

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

   if 7.8e8 < y

   1. Initial program 100.0%

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

      1. expm1-log1p-u 69.8%

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

   4. Applied egg-rr 69.8%

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

   5. Taylor expanded in y around 0 3.1%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \cos x\right)}\right) \]
   6. Step-by-step derivation

      1. log1p-define 3.1%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\cos x\right)}\right) \]

   7. Simplified 3.1%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\cos x\right)}\right) \]

   8. Taylor expanded in x around 0 12.0%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
   9. Step-by-step derivation

      1. *-commutative 12.0%

         \[\leadsto 1 + \color{blue}{{x}^{2} \cdot -0.5} \]

   10. Simplified 12.0%

       \[\leadsto \color{blue}{1 + {x}^{2} \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 51.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 47.1%

   \[\leadsto \color{blue}{\cos x} \]
4. Add Preprocessing

Alternative 4: 28.2% 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 100.0%

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

   1. expm1-log1p-u 87.8%

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

4. Applied egg-rr 87.8%

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

5. Taylor expanded in y around 0 46.8%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \cos x\right)}\right) \]
6. Step-by-step derivation

   1. log1p-define 47.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\cos x\right)}\right) \]

7. Simplified 47.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\cos x\right)}\right) \]

8. Taylor expanded in x around 0 23.4%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024087 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))