Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 6

Speedup: 1.0×

• 48.6% 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: 86.9% 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 98.9%

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

   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 75.9%

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

      1. *-un-lft-identity 75.9%

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

      2. add-log-exp 75.9%

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

      3. *-un-lft-identity 75.9%

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

      4. log-prod 75.9%

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

      5. metadata-eval 75.9%

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

      6. add-log-exp 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. +-lft-identity 75.9%

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

   7. Simplified 75.9%

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

Alternative 3: 56.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7600000:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7600000.0) (cos x) (/ 1.0 (/ y (+ y (* -0.5 (* y (* x x))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7600000.0) {
		tmp = cos(x);
	} else {
		tmp = 1.0 / (y / (y + (-0.5 * (y * (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 <= 7600000.0d0) then
        tmp = cos(x)
    else
        tmp = 1.0d0 / (y / (y + ((-0.5d0) * (y * (x * x)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7600000.0) {
		tmp = Math.cos(x);
	} else {
		tmp = 1.0 / (y / (y + (-0.5 * (y * (x * x)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7600000.0:
		tmp = math.cos(x)
	else:
		tmp = 1.0 / (y / (y + (-0.5 * (y * (x * x)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7600000.0)
		tmp = cos(x);
	else
		tmp = Float64(1.0 / Float64(y / Float64(y + Float64(-0.5 * Float64(y * Float64(x * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7600000.0)
		tmp = cos(x);
	else
		tmp = 1.0 / (y / (y + (-0.5 * (y * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7600000.0], N[Cos[x], $MachinePrecision], N[(1.0 / N[(y / N[(y + N[(-0.5 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.6e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 61.7%

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

   if 7.6e6 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. clear-num 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 3.1%

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

   6. Taylor expanded in x around 0 21.7%

      \[\leadsto \frac{1}{\frac{y}{\color{blue}{y + -0.5 \cdot \left({x}^{2} \cdot y\right)}}} \]
   7. Step-by-step derivation

      1. unpow2 21.7%

         \[\leadsto \frac{1}{\frac{y}{y + -0.5 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)}} \]

   8. Applied egg-rr 21.7%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7600000:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.2% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{y}{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (/ y (+ y (* -0.5 (* y (* x x)))))))

C

double code(double x, double y) {
	return 1.0 / (y / (y + (-0.5 * (y * (x * x)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / (y / (y + ((-0.5d0) * (y * (x * x)))))
end function

Java

public static double code(double x, double y) {
	return 1.0 / (y / (y + (-0.5 * (y * (x * x)))));
}

Python

def code(x, y):
	return 1.0 / (y / (y + (-0.5 * (y * (x * x)))))

Julia

function code(x, y)
	return Float64(1.0 / Float64(y / Float64(y + Float64(-0.5 * Float64(y * Float64(x * x))))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / (y / (y + (-0.5 * (y * (x * x)))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-*r/ 99.9%

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

   2. clear-num 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in y around 0 49.1%

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

6. Taylor expanded in x around 0 38.9%

   \[\leadsto \frac{1}{\frac{y}{\color{blue}{y + -0.5 \cdot \left({x}^{2} \cdot y\right)}}} \]
7. Step-by-step derivation

   1. unpow2 38.9%

      \[\leadsto \frac{1}{\frac{y}{y + -0.5 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)}} \]

8. Applied egg-rr 38.9%

   \[\leadsto \frac{1}{\frac{y}{y + -0.5 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)}} \]

9. Final simplification 38.9%

   \[\leadsto \frac{1}{\frac{y}{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}} \]
10. Add Preprocessing

Alternative 5: 33.3% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + -0.5 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* -0.5 (* x x))))

C

double code(double x, double y) {
	return 1.0 + (-0.5 * (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((-0.5d0) * (x * x))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (-0.5 * (x * x));
}

Python

def code(x, y):
	return 1.0 + (-0.5 * (x * x))

Julia

function code(x, y)
	return Float64(1.0 + Float64(-0.5 * Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (-0.5 * (x * x));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $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 49.2%

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

4. Taylor expanded in x around 0 34.5%

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

   1. *-commutative 34.5%

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

6. Simplified 34.5%

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

   1. unpow2 38.9%

      \[\leadsto \frac{1}{\frac{y}{y + -0.5 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)}} \]

8. Applied egg-rr 34.5%

   \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5 \]

9. Final simplification 34.5%

   \[\leadsto 1 + -0.5 \cdot \left(x \cdot x\right) \]
10. Add Preprocessing

Alternative 6: 29.3% 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 69.9%

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

4. Taylor expanded in y around 0 31.8%

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

Reproduce

herbie shell --seed 2024133 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))