Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 5

Speedup: 1.0×

• 49.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] / N[(y / N[Sinh[y], $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. clear-num 100.0%

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

   2. un-div-inv 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 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 3: 66.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+16}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.0003968253968253968 \cdot {y}^{6}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.2e+16)
   (cos x)
   (* 0.5 (+ 2.0 (* 0.0003968253968253968 (pow y 6.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.2e+16) {
		tmp = cos(x);
	} else {
		tmp = 0.5 * (2.0 + (0.0003968253968253968 * pow(y, 6.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.2d+16) then
        tmp = cos(x)
    else
        tmp = 0.5d0 * (2.0d0 + (0.0003968253968253968d0 * (y ** 6.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.2e+16) {
		tmp = Math.cos(x);
	} else {
		tmp = 0.5 * (2.0 + (0.0003968253968253968 * Math.pow(y, 6.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.2e+16:
		tmp = math.cos(x)
	else:
		tmp = 0.5 * (2.0 + (0.0003968253968253968 * math.pow(y, 6.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.2e+16)
		tmp = cos(x);
	else
		tmp = Float64(0.5 * Float64(2.0 + Float64(0.0003968253968253968 * (y ^ 6.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.2e+16)
		tmp = cos(x);
	else
		tmp = 0.5 * (2.0 + (0.0003968253968253968 * (y ^ 6.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.2e+16], N[Cos[x], $MachinePrecision], N[(0.5 * N[(2.0 + N[(0.0003968253968253968 * N[Power[y, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.2e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 64.7%

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

   if 5.2e16 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 77.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{y} - \frac{1}{e^{y}}}{y}} \]
   4. Step-by-step derivation

      1. rec-exp 77.4%

         \[\leadsto 0.5 \cdot \frac{e^{y} - \color{blue}{e^{-y}}}{y} \]

   5. Simplified 77.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{y} - e^{-y}}{y}} \]

   6. Taylor expanded in y around 0 65.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(2 + {y}^{2} \cdot \left(0.3333333333333333 + {y}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 65.3%

         \[\leadsto 0.5 \cdot \left(2 + {y}^{2} \cdot \left(0.3333333333333333 + {y}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{y}^{2} \cdot 0.0003968253968253968}\right)\right)\right) \]

   8. Simplified 65.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(2 + {y}^{2} \cdot \left(0.3333333333333333 + {y}^{2} \cdot \left(0.016666666666666666 + {y}^{2} \cdot 0.0003968253968253968\right)\right)\right)} \]

   9. Taylor expanded in y around inf 65.3%

      \[\leadsto 0.5 \cdot \left(2 + \color{blue}{0.0003968253968253968 \cdot {y}^{6}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+16}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + 0.0003968253968253968 \cdot {y}^{6}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.8% 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 49.8%

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

4. Final simplification 49.8%

   \[\leadsto \cos x \]
5. Add Preprocessing

Alternative 5: 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 40.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{e^{y} - \frac{1}{e^{y}}}{y}} \]
4. Step-by-step derivation

   1. rec-exp 40.5%

      \[\leadsto 0.5 \cdot \frac{e^{y} - \color{blue}{e^{-y}}}{y} \]

5. Simplified 40.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{e^{y} - e^{-y}}{y}} \]

6. Taylor expanded in y around 0 26.5%

   \[\leadsto 0.5 \cdot \color{blue}{2} \]

7. Final simplification 26.5%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))