Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 6.8s

Alternatives: 5

Speedup: 1.0×

• 50.7% 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: 99.9% 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: 99.9% 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 2: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t_0 \leq 1.00000000002:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.00000000002) (sin x) (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.00000000002) {
		tmp = sin(x);
	} else {
		tmp = x * 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 <= 1.00000000002d0) then
        tmp = sin(x)
    else
        tmp = x * 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 <= 1.00000000002) {
		tmp = Math.sin(x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if t_0 <= 1.00000000002:
		tmp = math.sin(x)
	else:
		tmp = x * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (t_0 <= 1.00000000002)
		tmp = sin(x);
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	tmp = 0.0;
	if (t_0 <= 1.00000000002)
		tmp = sin(x);
	else
		tmp = x * 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, 1.00000000002], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1.00000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.9%

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

   if 1.00000000002 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 74.6%

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

   3. Simplified 74.6%

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

   4. Taylor expanded in x around 0 54.7%

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

      1. associate-/r/ 80.1%

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

   6. Applied egg-rr 80.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 1.00000000002:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \end{array} \]

Alternative 3: 60.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.2e-5) (sin x) (+ x (* 0.16666666666666666 (* x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-5) {
		tmp = sin(x);
	} else {
		tmp = x + (0.16666666666666666 * (x * (y * 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 <= 1.2d-5) then
        tmp = sin(x)
    else
        tmp = x + (0.16666666666666666d0 * (x * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-5) {
		tmp = Math.sin(x);
	} else {
		tmp = x + (0.16666666666666666 * (x * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.2e-5:
		tmp = math.sin(x)
	else:
		tmp = x + (0.16666666666666666 * (x * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.2e-5)
		tmp = sin(x);
	else
		tmp = Float64(x + Float64(0.16666666666666666 * Float64(x * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.2e-5)
		tmp = sin(x);
	else
		tmp = x + (0.16666666666666666 * (x * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.2e-5], N[Sin[x], $MachinePrecision], N[(x + N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 70.1%

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

   if 1.2e-5 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in x around 0 53.7%

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

   5. Taylor expanded in y around 0 53.5%

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

      1. unpow2 53.5%

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

   7. Simplified 53.5%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 4: 48.1% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (* 0.16666666666666666 (* x (* y y)))))

C

double code(double x, double y) {
	return x + (0.16666666666666666 * (x * (y * y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (0.16666666666666666d0 * (x * (y * y)))
end function

Java

public static double code(double x, double y) {
	return x + (0.16666666666666666 * (x * (y * y)));
}

Python

def code(x, y):
	return x + (0.16666666666666666 * (x * (y * y)))

Julia

function code(x, y)
	return Float64(x + Float64(0.16666666666666666 * Float64(x * Float64(y * y))))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (0.16666666666666666 * (x * (y * y)));
end

Wolfram

code[x_, y_] := N[(x + N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-/r/ 87.4%

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

3. Simplified 87.4%

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

4. Taylor expanded in x around 0 53.5%

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

5. Taylor expanded in y around 0 52.4%

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

   1. unpow2 52.4%

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

7. Simplified 52.4%

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

8. Final simplification 52.4%

   \[\leadsto x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right) \]

Alternative 5: 26.4% 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. *-commutative 100.0%

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

   2. associate-/r/ 87.4%

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

3. Simplified 87.4%

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

4. Taylor expanded in y around 0 52.3%

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

5. Taylor expanded in x around 0 28.4%

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

6. Final simplification 28.4%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))