Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 15.7s

Alternatives: 8

Speedup: 1.0×

• Could not converge on a ground truth

  1. reg0 = (bf "1.289203255425954505063189166393795863163e-196")
  2. reg1 = (bf #e1.218305814288792410058393573407991292529e113)

• 50.2% 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: 84.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.012 \lor \neg \left(y \leq 1.4 \cdot 10^{+154}\right):\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 0.012) (not (<= y 1.4e+154)))
   (* (cos x) (+ 1.0 (* 0.16666666666666666 (* y y))))
   (/ (sinh y) y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 0.012) || !(y <= 1.4e+154)) {
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else {
		tmp = sinh(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 <= 0.012d0) .or. (.not. (y <= 1.4d+154))) then
        tmp = cos(x) * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    else
        tmp = sinh(y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 0.012) || !(y <= 1.4e+154)) {
		tmp = Math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else {
		tmp = Math.sinh(y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 0.012) or not (y <= 1.4e+154):
		tmp = math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)))
	else:
		tmp = math.sinh(y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 0.012) || !(y <= 1.4e+154))
		tmp = Float64(cos(x) * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	else
		tmp = Float64(sinh(y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 0.012) || ~((y <= 1.4e+154)))
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	else
		tmp = sinh(y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 0.012], N[Not[LessEqual[y, 1.4e+154]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.012 or 1.4e154 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 82.8%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. unpow2 82.8%

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

   5. Applied egg-rr 82.8%

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

   if 0.012 < y < 1.4e154

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 85.7%

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

      1. *-un-lft-identity 85.7%

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

      2. clear-num 85.7%

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

   5. Applied egg-rr 85.7%

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

      1. associate-/r/ 85.7%

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

      2. associate-*l/ 85.7%

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

      3. *-lft-identity 85.7%

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

   7. Simplified 85.7%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.012 \lor \neg \left(y \leq 1.4 \cdot 10^{+154}\right):\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00036:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.00036) (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.00036) {
		tmp = cos(x);
	} else {
		tmp = sinh(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 <= 0.00036d0) then
        tmp = cos(x)
    else
        tmp = sinh(y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.00036) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.sinh(y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.00036:
		tmp = math.cos(x)
	else:
		tmp = math.sinh(y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.00036)
		tmp = cos(x);
	else
		tmp = Float64(sinh(y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.00036)
		tmp = cos(x);
	else
		tmp = sinh(y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.00036], N[Cos[x], $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.60000000000000023e-4

   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 3.60000000000000023e-4 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. clear-num 82.9%

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

   5. Applied egg-rr 82.9%

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

      1. associate-/r/ 82.9%

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

      2. associate-*l/ 82.9%

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

      3. *-lft-identity 82.9%

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

   7. Simplified 82.9%

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

Alternative 4: 62.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 600:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+174}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot -0.5\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (<= y 600.0)
     (cos x)
     (if (<= y 9.5e+174) (* (+ 1.0 (* (* x x) -0.5)) t_0) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 600.0) {
		tmp = cos(x);
	} else if (y <= 9.5e+174) {
		tmp = (1.0 + ((x * x) * -0.5)) * t_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 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if (y <= 600.0d0) then
        tmp = cos(x)
    else if (y <= 9.5d+174) then
        tmp = (1.0d0 + ((x * x) * (-0.5d0))) * t_0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 600.0) {
		tmp = Math.cos(x);
	} else if (y <= 9.5e+174) {
		tmp = (1.0 + ((x * x) * -0.5)) * t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if y <= 600.0:
		tmp = math.cos(x)
	elif y <= 9.5e+174:
		tmp = (1.0 + ((x * x) * -0.5)) * t_0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (y <= 600.0)
		tmp = cos(x);
	elseif (y <= 9.5e+174)
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * -0.5)) * t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (y <= 600.0)
		tmp = cos(x);
	elseif (y <= 9.5e+174)
		tmp = (1.0 + ((x * x) * -0.5)) * t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 600.0], N[Cos[x], $MachinePrecision], If[LessEqual[y, 9.5e+174], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y < 600

   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 600 < y < 9.4999999999999992e174

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 12.3%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. unpow2 12.3%

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

   5. Applied egg-rr 12.3%

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

   6. Taylor expanded in x around 0 19.1%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative 19.1%

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot -0.5}\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   8. Simplified 19.1%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 19.1%

         \[\leadsto \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   10. Applied egg-rr 19.1%

       \[\leadsto \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   if 9.4999999999999992e174 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 81.3%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 600:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+174}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot -0.5\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;x \leq 7.5 \cdot 10^{+123}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot -0.5\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (<= x 7.5e+123) t_0 (* (+ 1.0 (* (* x x) -0.5)) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (x <= 7.5e+123) {
		tmp = t_0;
	} else {
		tmp = (1.0 + ((x * x) * -0.5)) * 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 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if (x <= 7.5d+123) then
        tmp = t_0
    else
        tmp = (1.0d0 + ((x * x) * (-0.5d0))) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (x <= 7.5e+123) {
		tmp = t_0;
	} else {
		tmp = (1.0 + ((x * x) * -0.5)) * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if x <= 7.5e+123:
		tmp = t_0
	else:
		tmp = (1.0 + ((x * x) * -0.5)) * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (x <= 7.5e+123)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * -0.5)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (x <= 7.5e+123)
		tmp = t_0;
	else
		tmp = (1.0 + ((x * x) * -0.5)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 7.5e+123], t$95$0, N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.4999999999999999e123

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 72.8%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. unpow2 72.8%

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

   5. Applied egg-rr 72.8%

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

   6. Taylor expanded in x around 0 53.6%

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

   if 7.4999999999999999e123 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 68.3%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. unpow2 68.3%

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

   5. Applied egg-rr 68.3%

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

   6. Taylor expanded in x around 0 39.8%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative 39.8%

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot -0.5}\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   8. Simplified 39.8%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 39.8%

         \[\leadsto \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   10. Applied egg-rr 39.8%

       \[\leadsto \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+123}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot -0.5\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.9% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{+154}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.6e+154)
   (+ 1.0 (* 0.16666666666666666 (* y y)))
   (+ 1.0 (* (* x x) -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.6e+154) {
		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 (x <= 6.6d+154) 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 (x <= 6.6e+154) {
		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 x <= 6.6e+154:
		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 (x <= 6.6e+154)
		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 (x <= 6.6e+154)
		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[LessEqual[x, 6.6e+154], 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 x < 6.6e154

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 71.9%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. unpow2 71.9%

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

   5. Applied egg-rr 71.9%

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

   6. Taylor expanded in x around 0 52.1%

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

   if 6.6e154 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 36.5%

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

   4. Taylor expanded in x around 0 42.2%

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

      1. *-commutative 42.2%

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot -0.5}\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   6. Simplified 42.2%

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

      1. unpow2 42.2%

         \[\leadsto \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   8. Applied egg-rr 42.2%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{+154}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.1% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot x\right) \cdot -0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* (* x x) -0.5)))

C

double code(double x, double y) {
	return 1.0 + ((x * x) * -0.5);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((x * x) * (-0.5d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((x * x) * -0.5);
}

Python

def code(x, y):
	return 1.0 + ((x * x) * -0.5)

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(x * x) * -0.5))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((x * x) * -0.5);
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 45.1%

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

4. Taylor expanded in x around 0 32.3%

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

   1. *-commutative 49.2%

      \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot -0.5}\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

6. Simplified 32.3%

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

   1. unpow2 49.2%

      \[\leadsto \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

8. Applied egg-rr 32.3%

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

Alternative 8: 28.2% 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 x around 0 70.7%

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

4. Taylor expanded in y around 0 27.3%

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

Reproduce

herbie shell --seed 2024143 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))