Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 5

Speedup: 1.0×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Step-by-step derivation

   1. clear-num 100.0%

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

   2. un-div-inv 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \frac{\sin x}{\frac{y}{\sinh y}} \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]

2. Final simplification 100.0%

   \[\leadsto \sin x \cdot \frac{\sinh y}{y} \]

Alternative 3: 61.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-15}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.2e-15) (sin x) (fma 0.16666666666666666 (* x (* y y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-15) {
		tmp = sin(x);
	} else {
		tmp = fma(0.16666666666666666, (x * (y * y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.2e-15)
		tmp = sin(x);
	else
		tmp = fma(0.16666666666666666, Float64(x * Float64(y * y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.2e-15], N[Sin[x], $MachinePrecision], N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.19999999999999986e-15

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 65.4%

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

   if 2.19999999999999986e-15 < y

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. clear-num 100.0%

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

      2. un-div-inv 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around 0 4.5%

      \[\leadsto \frac{\sin x}{\color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 4.5%

         \[\leadsto \frac{\sin x}{1 + -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}} \]

   6. Simplified 4.5%

      \[\leadsto \frac{\sin x}{\color{blue}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}} \]

   7. Taylor expanded in x around 0 4.7%

      \[\leadsto \color{blue}{\frac{x}{1 + -0.16666666666666666 \cdot {y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 4.7%

         \[\leadsto \frac{x}{1 + -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}} \]

   9. Simplified 4.7%

      \[\leadsto \color{blue}{\frac{x}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}} \]

   10. Taylor expanded in y around 0 54.6%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({y}^{2} \cdot x\right) + x} \]
   11. Step-by-step derivation

       1. fma-def 54.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, {y}^{2} \cdot x, x\right)} \]

       2. *-commutative 54.6%

          \[\leadsto \mathsf{fma}\left(0.16666666666666666, \color{blue}{x \cdot {y}^{2}}, x\right) \]

       3. unpow2 54.6%

          \[\leadsto \mathsf{fma}\left(0.16666666666666666, x \cdot \color{blue}{\left(y \cdot y\right)}, x\right) \]

   12. Simplified 54.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-15}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \]

Alternative 4: 50.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]

2. Taylor expanded in y around 0 50.2%

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

3. Final simplification 50.2%

   \[\leadsto \sin x \]

Alternative 5: 26.5% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Step-by-step derivation

   1. clear-num 100.0%

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

   2. un-div-inv 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in y around 0 49.6%

   \[\leadsto \frac{\sin x}{\color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 49.6%

      \[\leadsto \frac{\sin x}{1 + -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}} \]

6. Simplified 49.6%

   \[\leadsto \frac{\sin x}{\color{blue}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}} \]

7. Taylor expanded in x around 0 24.2%

   \[\leadsto \color{blue}{\frac{x}{1 + -0.16666666666666666 \cdot {y}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 24.2%

      \[\leadsto \frac{x}{1 + -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}} \]

9. Simplified 24.2%

   \[\leadsto \color{blue}{\frac{x}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}} \]

10. Taylor expanded in y around 0 24.6%

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

11. Final simplification 24.6%

    \[\leadsto x \]

Reproduce

herbie shell --seed 2023257 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))