Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 10.6s

Alternatives: 5

Speedup: 1.0×

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

Specification

Math

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

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. Final simplification 100.0%

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

Alternative 2: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;\frac{\cos x}{y \cdot \left(y \cdot -0.16666666666666666\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= t_0 2.0) (/ (cos x) (+ (* y (* y -0.16666666666666666)) 1.0)) 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) / ((y * (y * -0.16666666666666666)) + 1.0);
	} 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) / ((y * (y * (-0.16666666666666666d0))) + 1.0d0)
    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) / ((y * (y * -0.16666666666666666)) + 1.0);
	} 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) / ((y * (y * -0.16666666666666666)) + 1.0)
	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 = Float64(cos(x) / Float64(Float64(y * Float64(y * -0.16666666666666666)) + 1.0));
	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) / ((y * (y * -0.16666666666666666)) + 1.0);
	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[(N[Cos[x], $MachinePrecision] / N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos x \cdot \frac{\sinh y}{y}\right)\right)} \]

      2. expm1-udef 99.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos x \cdot \frac{\sinh y}{y}\right)} - 1} \]

      3. sub-neg 99.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos x \cdot \frac{\sinh y}{y}\right)} + \left(-1\right)} \]

      4. *-commutative 99.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sinh y}{y} \cdot \cos x}\right)} + \left(-1\right) \]

      5. div-inv 99.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sinh y \cdot \frac{1}{y}\right)} \cdot \cos x\right)} + \left(-1\right) \]

      6. associate-*l* 99.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sinh y \cdot \left(\frac{1}{y} \cdot \cos x\right)}\right)} + \left(-1\right) \]

      7. associate-*l/ 99.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sinh y \cdot \color{blue}{\frac{1 \cdot \cos x}{y}}\right)} + \left(-1\right) \]

      8. *-un-lft-identity 99.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sinh y \cdot \frac{\color{blue}{\cos x}}{y}\right)} + \left(-1\right) \]

      9. metadata-eval 99.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sinh y \cdot \frac{\cos x}{y}\right)} + \color{blue}{-1} \]

   3. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y \cdot \frac{\cos x}{y}\right)} + -1} \]
   4. Step-by-step derivation

      1. metadata-eval 99.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sinh y \cdot \frac{\cos x}{y}\right)} + \color{blue}{\left(-1\right)} \]

      2. sub-neg 99.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sinh y \cdot \frac{\cos x}{y}\right)} - 1} \]

      3. expm1-def 99.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh y \cdot \frac{\cos x}{y}\right)\right)} \]

      4. expm1-log1p 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-/r/ 99.9%

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

   5. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. div-inv 99.7%

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

      3. times-frac 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in y around 0 99.3%

      \[\leadsto \frac{1}{y} \cdot \frac{\cos x}{\color{blue}{-0.16666666666666666 \cdot y + \frac{1}{y}}} \]

   9. Taylor expanded in x around inf 99.3%

      \[\leadsto \color{blue}{\frac{\cos x}{y \cdot \left(-0.16666666666666666 \cdot y + \frac{1}{y}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 99.3%

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

       2. distribute-lft-in 99.3%

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

       3. *-commutative 99.3%

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

       4. lft-mult-inverse 99.5%

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

   11. Simplified 99.5%

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

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

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/l* 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 76.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 2:\\ \;\;\;\;\frac{\cos x}{y \cdot \left(y \cdot -0.16666666666666666\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]

Alternative 3: 86.3% 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 99.9%

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

   2. Taylor expanded in y around 0 98.7%

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

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

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/l* 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 76.0%

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

4. Final simplification 87.3%

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

Alternative 4: 51.0% 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. Taylor expanded in y around 0 50.6%

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

3. Final simplification 50.6%

   \[\leadsto \cos x \]

Alternative 5: 28.3% 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. Taylor expanded in y around 0 50.6%

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

3. Taylor expanded in x around 0 27.0%

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

4. Final simplification 27.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))