Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 6

Speedup: 1.0×

• 48.4% 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: 56.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 155000000000:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{4} \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 155000000000.0) (cos x) (sqrt (* (pow x 4.0) 0.25))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 155000000000.0) {
		tmp = cos(x);
	} else {
		tmp = sqrt((pow(x, 4.0) * 0.25));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 155000000000.0d0) then
        tmp = cos(x)
    else
        tmp = sqrt(((x ** 4.0d0) * 0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 155000000000.0) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.sqrt((Math.pow(x, 4.0) * 0.25));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 155000000000.0:
		tmp = math.cos(x)
	else:
		tmp = math.sqrt((math.pow(x, 4.0) * 0.25))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 155000000000.0)
		tmp = cos(x);
	else
		tmp = sqrt(Float64((x ^ 4.0) * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 155000000000.0)
		tmp = cos(x);
	else
		tmp = sqrt(((x ^ 4.0) * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 155000000000.0], N[Cos[x], $MachinePrecision], N[Sqrt[N[(N[Power[x, 4.0], $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55e11

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 69.0%

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

   if 1.55e11 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 3.1%

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

   4. Taylor expanded in x around 0 16.1%

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

   5. Taylor expanded in x around inf 15.3%

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

      1. add-sqr-sqrt 0.3%

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

      2. sqrt-unprod 22.0%

         \[\leadsto \color{blue}{\sqrt{\left(-0.5 \cdot {x}^{2}\right) \cdot \left(-0.5 \cdot {x}^{2}\right)}} \]

      3. pow1/2 22.0%

         \[\leadsto \color{blue}{{\left(\left(-0.5 \cdot {x}^{2}\right) \cdot \left(-0.5 \cdot {x}^{2}\right)\right)}^{0.5}} \]

      4. *-commutative 22.0%

         \[\leadsto {\left(\color{blue}{\left({x}^{2} \cdot -0.5\right)} \cdot \left(-0.5 \cdot {x}^{2}\right)\right)}^{0.5} \]

      5. *-commutative 22.0%

         \[\leadsto {\left(\left({x}^{2} \cdot -0.5\right) \cdot \color{blue}{\left({x}^{2} \cdot -0.5\right)}\right)}^{0.5} \]

      6. swap-sqr 22.0%

         \[\leadsto {\color{blue}{\left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(-0.5 \cdot -0.5\right)\right)}}^{0.5} \]

      7. pow-prod-up 22.0%

         \[\leadsto {\left(\color{blue}{{x}^{\left(2 + 2\right)}} \cdot \left(-0.5 \cdot -0.5\right)\right)}^{0.5} \]

      8. metadata-eval 22.0%

         \[\leadsto {\left({x}^{\color{blue}{4}} \cdot \left(-0.5 \cdot -0.5\right)\right)}^{0.5} \]

      9. metadata-eval 22.0%

         \[\leadsto {\left({x}^{4} \cdot \color{blue}{0.25}\right)}^{0.5} \]

   7. Applied egg-rr 22.0%

      \[\leadsto \color{blue}{{\left({x}^{4} \cdot 0.25\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 22.0%

         \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 0.25}} \]

   9. Simplified 22.0%

      \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 0.25}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 55.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 80000000:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 80000000.0) (cos x) (+ 1.0 (* -0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 80000000.0) {
		tmp = cos(x);
	} else {
		tmp = 1.0 + (-0.5 * (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 <= 80000000.0d0) then
        tmp = cos(x)
    else
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 80000000.0) {
		tmp = Math.cos(x);
	} else {
		tmp = 1.0 + (-0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 80000000.0:
		tmp = math.cos(x)
	else:
		tmp = 1.0 + (-0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 80000000.0)
		tmp = cos(x);
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 80000000.0)
		tmp = cos(x);
	else
		tmp = 1.0 + (-0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 80000000.0], N[Cos[x], $MachinePrecision], N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8e7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 69.7%

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

   if 8e7 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 3.1%

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

   4. Taylor expanded in x around 0 17.1%

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

      1. unpow2 17.1%

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

   6. Applied egg-rr 17.1%

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

Alternative 4: 32.1% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 80000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 80000000.0) 1.0 (+ 1.0 (* -0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 80000000.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (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 <= 80000000.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 80000000.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 80000000.0:
		tmp = 1.0
	else:
		tmp = 1.0 + (-0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 80000000.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 80000000.0)
		tmp = 1.0;
	else
		tmp = 1.0 + (-0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 80000000.0], 1.0, N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8e7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 69.7%

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

   4. Taylor expanded in x around 0 37.4%

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

   5. Taylor expanded in x around 0 37.7%

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

   if 8e7 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 3.1%

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

   4. Taylor expanded in x around 0 17.1%

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

      1. unpow2 17.1%

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

   6. Applied egg-rr 17.1%

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

Alternative 5: 31.9% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 300000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 300000000.0) 1.0 (* -0.5 (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 300000000.0) {
		tmp = 1.0;
	} else {
		tmp = -0.5 * (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 <= 300000000.0d0) then
        tmp = 1.0d0
    else
        tmp = (-0.5d0) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 300000000.0) {
		tmp = 1.0;
	} else {
		tmp = -0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 300000000.0:
		tmp = 1.0
	else:
		tmp = -0.5 * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 300000000.0)
		tmp = 1.0;
	else
		tmp = Float64(-0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 300000000.0)
		tmp = 1.0;
	else
		tmp = -0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 300000000.0], 1.0, N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3e8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 69.7%

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

   4. Taylor expanded in x around 0 37.4%

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

   5. Taylor expanded in x around 0 37.7%

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

   if 3e8 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 3.1%

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

   4. Taylor expanded in x around 0 17.1%

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

   5. Taylor expanded in x around inf 16.2%

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

      1. unpow2 17.1%

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

   7. Applied egg-rr 16.2%

      \[\leadsto -0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 29.2% 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 y around 0 50.5%

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

4. Taylor expanded in x around 0 31.5%

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

5. Taylor expanded in x around 0 27.5%

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

Reproduce

herbie shell --seed 2024116 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))