Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 6

Speedup: 1.0×

• 49.6% 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: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 2.0) (cos x) t_0)))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 2.0) {
		tmp = cos(x);
	} 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 = sinh(y) / y
    if (t_0 <= 2.0d0) then
        tmp = cos(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(y) / y;
	double tmp;
	if (t_0 <= 2.0) {
		tmp = Math.cos(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if t_0 <= 2.0:
		tmp = math.cos(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = cos(x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = cos(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], N[Cos[x], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.8%

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

   if 2 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 67.0%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 67.0%

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

      2. add-log-exp 66.1%

         \[\leadsto \color{blue}{\log \left(e^{\frac{\sinh y}{y}}\right)} \]

      3. *-un-lft-identity 66.1%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\frac{\sinh y}{y}}\right)} \]

      4. log-prod 66.1%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\frac{\sinh y}{y}}\right)} \]

      5. metadata-eval 66.1%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\sinh y}{y}}\right) \]

      6. add-log-exp 67.0%

         \[\leadsto 0 + \color{blue}{\frac{\sinh y}{y}} \]

   5. Applied egg-rr 67.0%

      \[\leadsto \color{blue}{0 + \frac{\sinh y}{y}} \]
   6. Step-by-step derivation

      1. +-lft-identity 67.0%

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

   7. Simplified 67.0%

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

Alternative 3: 55.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+38}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.15e+38) (cos x) (/ (+ y (* -0.5 (* y (* x x)))) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.15e+38) {
		tmp = cos(x);
	} else {
		tmp = (y + (-0.5 * (y * (x * x)))) / 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 <= 2.15d+38) then
        tmp = cos(x)
    else
        tmp = (y + ((-0.5d0) * (y * (x * x)))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.15e+38) {
		tmp = Math.cos(x);
	} else {
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.15e+38:
		tmp = math.cos(x)
	else:
		tmp = (y + (-0.5 * (y * (x * x)))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.15e+38)
		tmp = cos(x);
	else
		tmp = Float64(Float64(y + Float64(-0.5 * Float64(y * Float64(x * x)))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.15e+38)
		tmp = cos(x);
	else
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.15e+38], N[Cos[x], $MachinePrecision], N[(N[(y + N[(-0.5 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1499999999999998e38

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 72.5%

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

   if 2.1499999999999998e38 < y

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 100.0%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \cos x \cdot \frac{\sinh y}{y}} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 3.1%

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

   8. Taylor expanded in x around 0 36.7%

      \[\leadsto \frac{\color{blue}{y + -0.5 \cdot \left({x}^{2} \cdot y\right)}}{y} \]
   9. Step-by-step derivation

      1. unpow2 36.7%

         \[\leadsto \frac{y + -0.5 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)}{y} \]

   10. Applied egg-rr 36.7%

       \[\leadsto \frac{y + -0.5 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+38}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 33.5% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.6e-24) 1.0 (/ (+ y (* -0.5 (* y (* x x)))) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-24) {
		tmp = 1.0;
	} else {
		tmp = (y + (-0.5 * (y * (x * x)))) / 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 <= 2.6d-24) then
        tmp = 1.0d0
    else
        tmp = (y + ((-0.5d0) * (y * (x * x)))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-24) {
		tmp = 1.0;
	} else {
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.6e-24:
		tmp = 1.0
	else:
		tmp = (y + (-0.5 * (y * (x * x)))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e-24)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y + Float64(-0.5 * Float64(y * Float64(x * x)))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.6e-24)
		tmp = 1.0;
	else
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e-24], 1.0, N[(N[(y + N[(-0.5 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6e-24

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 61.2%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   4. Taylor expanded in y around 0 42.6%

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

   if 2.6e-24 < y

   1. Initial program 100.0%

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

      1. add-log-exp 98.5%

         \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 98.5%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 98.5%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 98.5%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cos x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \cos x \cdot \frac{\sinh y}{y}} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 12.1%

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

   8. Taylor expanded in x around 0 37.3%

      \[\leadsto \frac{\color{blue}{y + -0.5 \cdot \left({x}^{2} \cdot y\right)}}{y} \]
   9. Step-by-step derivation

      1. unpow2 37.3%

         \[\leadsto \frac{y + -0.5 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)}{y} \]

   10. Applied egg-rr 37.3%

       \[\leadsto \frac{y + -0.5 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.1% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.6e-24) 1.0 (+ 1.0 (* -0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-24) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.6d-24) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-24) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.6e-24:
		tmp = 1.0
	else:
		tmp = 1.0 + (-0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e-24)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.6e-24)
		tmp = 1.0;
	else
		tmp = 1.0 + (-0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e-24], 1.0, N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6e-24

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 61.2%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   4. Taylor expanded in y around 0 42.6%

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

   if 2.6e-24 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 12.1%

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

   4. Taylor expanded in x around 0 24.1%

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

      1. *-commutative 24.1%

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

   6. Simplified 24.1%

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

      1. unpow2 37.3%

         \[\leadsto \frac{y + -0.5 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)}{y} \]

   8. Applied egg-rr 24.1%

      \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 28.9% 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. Taylor expanded in x around 0 61.9%

   \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

4. Taylor expanded in y around 0 34.4%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))