Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 6

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 9.118258959552568e-77
  2. y = 8.901822639768068e+66

• 49.5% 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: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 1:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.0) (cos x) t_0)))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		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 <= 1.0:
		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 <= 1.0)
		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 <= 1.0)
		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, 1.0], N[Cos[x], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   if 1 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 71.6%

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

      1. *-un-lft-identity 71.6%

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

      2. add-log-exp 70.8%

         \[\leadsto \color{blue}{\log \left(e^{\frac{\sinh y}{y}}\right)} \]

      3. *-un-lft-identity 70.8%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\frac{\sinh y}{y}}\right)} \]

      4. log-prod 70.8%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\frac{\sinh y}{y}}\right)} \]

      5. metadata-eval 70.8%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\sinh y}{y}}\right) \]

      6. add-log-exp 71.6%

         \[\leadsto 0 + \color{blue}{\frac{\sinh y}{y}} \]

   5. Applied egg-rr 71.6%

      \[\leadsto \color{blue}{0 + \frac{\sinh y}{y}} \]
   6. Step-by-step derivation

      1. +-lft-identity 71.6%

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

   7. Simplified 71.6%

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

Alternative 3: 61.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 18500000:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+140}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 18500000.0)
   (cos x)
   (if (<= y 6e+140)
     (+ 1.0 (* (* x x) -0.5))
     (+ 1.0 (* 0.16666666666666666 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 18500000.0) {
		tmp = cos(x);
	} else if (y <= 6e+140) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else {
		tmp = 1.0 + (0.16666666666666666 * (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 <= 18500000.0d0) then
        tmp = cos(x)
    else if (y <= 6d+140) then
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    else
        tmp = 1.0d0 + (0.16666666666666666d0 * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 18500000.0) {
		tmp = Math.cos(x);
	} else if (y <= 6e+140) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 18500000.0:
		tmp = math.cos(x)
	elif y <= 6e+140:
		tmp = 1.0 + ((x * x) * -0.5)
	else:
		tmp = 1.0 + (0.16666666666666666 * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 18500000.0)
		tmp = cos(x);
	elseif (y <= 6e+140)
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	else
		tmp = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 18500000.0)
		tmp = cos(x);
	elseif (y <= 6e+140)
		tmp = 1.0 + ((x * x) * -0.5);
	else
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 18500000.0], N[Cos[x], $MachinePrecision], If[LessEqual[y, 6e+140], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.85e7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 60.9%

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

   if 1.85e7 < y < 5.99999999999999993e140

   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 30.0%

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

      1. *-commutative 30.0%

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

   6. Simplified 30.0%

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

      1. unpow2 30.0%

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

   8. Applied egg-rr 30.0%

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

   if 5.99999999999999993e140 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 80.6%

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

   4. Taylor expanded in y around 0 73.2%

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

      1. unpow2 73.2%

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

   6. Applied egg-rr 73.2%

      \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 48.6% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1050000000 \lor \neg \left(y \leq 6 \cdot 10^{+140}\right):\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 1050000000.0) (not (<= y 6e+140)))
   (+ 1.0 (* 0.16666666666666666 (* y y)))
   (+ 1.0 (* (* x x) -0.5))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 1050000000.0) || !(y <= 6e+140)) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else {
		tmp = 1.0 + ((x * x) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 1050000000.0d0) .or. (.not. (y <= 6d+140))) then
        tmp = 1.0d0 + (0.16666666666666666d0 * (y * y))
    else
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 1050000000.0) || !(y <= 6e+140)) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else {
		tmp = 1.0 + ((x * x) * -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 1050000000.0) or not (y <= 6e+140):
		tmp = 1.0 + (0.16666666666666666 * (y * y))
	else:
		tmp = 1.0 + ((x * x) * -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 1050000000.0) || !(y <= 6e+140))
		tmp = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)));
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 1050000000.0) || ~((y <= 6e+140)))
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	else
		tmp = 1.0 + ((x * x) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 1050000000.0], N[Not[LessEqual[y, 6e+140]], $MachinePrecision]], N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e9 or 5.99999999999999993e140 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 65.2%

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

   4. Taylor expanded in y around 0 50.5%

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

      1. unpow2 50.5%

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

   6. Applied egg-rr 50.5%

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

   if 1.05e9 < y < 5.99999999999999993e140

   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 31.1%

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

      1. *-commutative 31.1%

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

   6. Simplified 31.1%

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

      1. unpow2 31.1%

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

   8. Applied egg-rr 31.1%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1050000000 \lor \neg \left(y \leq 6 \cdot 10^{+140}\right):\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.4% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + 0.16666666666666666 \cdot \left(y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* 0.16666666666666666 (* y y))))

C

double code(double x, double y) {
	return 1.0 + (0.16666666666666666 * (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (0.16666666666666666d0 * (y * y))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (0.16666666666666666 * (y * y));
}

Python

def code(x, y):
	return 1.0 + (0.16666666666666666 * (y * y))

Julia

function code(x, y)
	return Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (0.16666666666666666 * (y * y));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 64.6%

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

4. Taylor expanded in y around 0 46.1%

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

   1. unpow2 46.1%

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

6. Applied egg-rr 46.1%

   \[\leadsto 1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]
7. Add Preprocessing

Alternative 6: 29.3% 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. Add Preprocessing

3. Taylor expanded in x around 0 64.6%

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

4. Taylor expanded in y around 0 27.2%

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

Reproduce

herbie shell --seed 2024172 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))