Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 7

Speedup: 1.0×

• 50.5% 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. Final simplification 100.0%

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

Alternative 2: 52.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 65000000000000:\\ \;\;\;\;\cos x \cdot \frac{y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{16 + -8 \cdot {x}^{2}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 65000000000000.0)
   (* (cos x) (/ y y))
   (/ (+ 16.0 (* -8.0 (pow x 2.0))) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 65000000000000.0) {
		tmp = cos(x) * (y / y);
	} else {
		tmp = (16.0 + (-8.0 * pow(x, 2.0))) / 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 <= 65000000000000.0d0) then
        tmp = cos(x) * (y / y)
    else
        tmp = (16.0d0 + ((-8.0d0) * (x ** 2.0d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 65000000000000.0) {
		tmp = Math.cos(x) * (y / y);
	} else {
		tmp = (16.0 + (-8.0 * Math.pow(x, 2.0))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 65000000000000.0:
		tmp = math.cos(x) * (y / y)
	else:
		tmp = (16.0 + (-8.0 * math.pow(x, 2.0))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 65000000000000.0)
		tmp = Float64(cos(x) * Float64(y / y));
	else
		tmp = Float64(Float64(16.0 + Float64(-8.0 * (x ^ 2.0))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 65000000000000.0)
		tmp = cos(x) * (y / y);
	else
		tmp = (16.0 + (-8.0 * (x ^ 2.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 65000000000000.0], N[(N[Cos[x], $MachinePrecision] * N[(y / y), $MachinePrecision]), $MachinePrecision], N[(N[(16.0 + N[(-8.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5e13

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.2%

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

   if 6.5e13 < y

   1. Initial program 100.0%

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

   4. Applied egg-rr 2.2%

      \[\leadsto \cos x \cdot \frac{\color{blue}{16}}{y} \]

   5. Taylor expanded in x around 0 12.5%

      \[\leadsto \color{blue}{-8 \cdot \frac{{x}^{2}}{y} + 16 \cdot \frac{1}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 12.5%

         \[\leadsto \color{blue}{\frac{-8 \cdot {x}^{2}}{y}} + 16 \cdot \frac{1}{y} \]

      2. *-rgt-identity 12.5%

         \[\leadsto \frac{\color{blue}{\left(-8 \cdot {x}^{2}\right) \cdot 1}}{y} + 16 \cdot \frac{1}{y} \]

      3. associate-*r/ 12.5%

         \[\leadsto \color{blue}{\left(-8 \cdot {x}^{2}\right) \cdot \frac{1}{y}} + 16 \cdot \frac{1}{y} \]

      4. distribute-rgt-out 12.5%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \left(-8 \cdot {x}^{2} + 16\right)} \]

      5. associate-*l/ 12.5%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(-8 \cdot {x}^{2} + 16\right)}{y}} \]

      6. *-lft-identity 12.5%

         \[\leadsto \frac{\color{blue}{-8 \cdot {x}^{2} + 16}}{y} \]

      7. +-commutative 12.5%

         \[\leadsto \frac{\color{blue}{16 + -8 \cdot {x}^{2}}}{y} \]

   7. Simplified 12.5%

      \[\leadsto \color{blue}{\frac{16 + -8 \cdot {x}^{2}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 65000000000000:\\ \;\;\;\;\cos x \cdot \frac{y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{16 + -8 \cdot {x}^{2}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+14}:\\ \;\;\;\;\cos x \cdot \frac{y}{y}\\ \mathbf{else}:\\ \;\;\;\;-8 \cdot \frac{{x}^{2}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.5e+14) (* (cos x) (/ y y)) (* -8.0 (/ (pow x 2.0) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.5e+14) {
		tmp = cos(x) * (y / y);
	} else {
		tmp = -8.0 * (pow(x, 2.0) / 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 <= 7.5d+14) then
        tmp = cos(x) * (y / y)
    else
        tmp = (-8.0d0) * ((x ** 2.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.5e+14) {
		tmp = Math.cos(x) * (y / y);
	} else {
		tmp = -8.0 * (Math.pow(x, 2.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.5e+14:
		tmp = math.cos(x) * (y / y)
	else:
		tmp = -8.0 * (math.pow(x, 2.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.5e+14)
		tmp = Float64(cos(x) * Float64(y / y));
	else
		tmp = Float64(-8.0 * Float64((x ^ 2.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.5e+14)
		tmp = cos(x) * (y / y);
	else
		tmp = -8.0 * ((x ^ 2.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.5e+14], N[(N[Cos[x], $MachinePrecision] * N[(y / y), $MachinePrecision]), $MachinePrecision], N[(-8.0 * N[(N[Power[x, 2.0], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5e14

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.2%

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

   if 7.5e14 < y

   1. Initial program 100.0%

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

   4. Applied egg-rr 2.2%

      \[\leadsto \cos x \cdot \frac{\color{blue}{16}}{y} \]

   5. Taylor expanded in x around 0 12.5%

      \[\leadsto \color{blue}{-8 \cdot \frac{{x}^{2}}{y} + 16 \cdot \frac{1}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 12.5%

         \[\leadsto \color{blue}{\frac{-8 \cdot {x}^{2}}{y}} + 16 \cdot \frac{1}{y} \]

      2. *-rgt-identity 12.5%

         \[\leadsto \frac{\color{blue}{\left(-8 \cdot {x}^{2}\right) \cdot 1}}{y} + 16 \cdot \frac{1}{y} \]

      3. associate-*r/ 12.5%

         \[\leadsto \color{blue}{\left(-8 \cdot {x}^{2}\right) \cdot \frac{1}{y}} + 16 \cdot \frac{1}{y} \]

      4. distribute-rgt-out 12.5%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \left(-8 \cdot {x}^{2} + 16\right)} \]

      5. associate-*l/ 12.5%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(-8 \cdot {x}^{2} + 16\right)}{y}} \]

      6. *-lft-identity 12.5%

         \[\leadsto \frac{\color{blue}{-8 \cdot {x}^{2} + 16}}{y} \]

      7. +-commutative 12.5%

         \[\leadsto \frac{\color{blue}{16 + -8 \cdot {x}^{2}}}{y} \]

   7. Simplified 12.5%

      \[\leadsto \color{blue}{\frac{16 + -8 \cdot {x}^{2}}{y}} \]

   8. Taylor expanded in x around inf 12.0%

      \[\leadsto \color{blue}{-8 \cdot \frac{{x}^{2}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+14}:\\ \;\;\;\;\cos x \cdot \frac{y}{y}\\ \mathbf{else}:\\ \;\;\;\;-8 \cdot \frac{{x}^{2}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 11.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (cos x) -5e-310) -2.0 1.5))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -5e-310) {
		tmp = -2.0;
	} else {
		tmp = 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (cos(x) <= (-5d-310)) then
        tmp = -2.0d0
    else
        tmp = 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.cos(x) <= -5e-310) {
		tmp = -2.0;
	} else {
		tmp = 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cos(x) <= -5e-310:
		tmp = -2.0
	else:
		tmp = 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -5e-310)
		tmp = -2.0;
	else
		tmp = 1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (cos(x) <= -5e-310)
		tmp = -2.0;
	else
		tmp = 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -5e-310], -2.0, 1.5]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -4.999999999999985e-310

   1. Initial program 100.0%

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

   4. Applied egg-rr 3.0%

      \[\leadsto \cos x \cdot \frac{\color{blue}{16}}{y} \]

   6. Applied egg-rr 11.1%

      \[\leadsto \color{blue}{-2} \]

   if -4.999999999999985e-310 < (cos.f64 x)

   1. Initial program 100.0%

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

   4. Applied egg-rr 2.4%

      \[\leadsto \cos x \cdot \frac{\color{blue}{16}}{y} \]

   6. Applied egg-rr 11.3%

      \[\leadsto \color{blue}{1.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 51.6%

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

4. Final simplification 51.6%

   \[\leadsto \cos x \cdot \frac{y}{y} \]
5. Add Preprocessing

Alternative 6: 3.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -2.0)

C

double code(double x, double y) {
	return -2.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -2.0d0
end function

Java

public static double code(double x, double y) {
	return -2.0;
}

Python

def code(x, y):
	return -2.0

Julia

function code(x, y)
	return -2.0
end

MATLAB

function tmp = code(x, y)
	tmp = -2.0;
end

Wolfram

code[x_, y_] := -2.0

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 2.5%

   \[\leadsto \cos x \cdot \frac{\color{blue}{16}}{y} \]

6. Applied egg-rr 3.5%

   \[\leadsto \color{blue}{-2} \]

7. Final simplification 3.5%

   \[\leadsto -2 \]
8. Add Preprocessing

Alternative 7: 8.6% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.5)

C

double code(double x, double y) {
	return 0.5;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0
end function

Java

public static double code(double x, double y) {
	return 0.5;
}

Python

def code(x, y):
	return 0.5

Julia

function code(x, y)
	return 0.5
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5;
end

Wolfram

code[x_, y_] := 0.5

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 2.5%

   \[\leadsto \cos x \cdot \frac{\color{blue}{16}}{y} \]

6. Applied egg-rr 8.5%

   \[\leadsto \color{blue}{0.5} \]

7. Final simplification 8.5%

   \[\leadsto 0.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))