Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

Alternatives: 6

Speedup: 1.0×

• 50.3% 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: 100.0% 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: 100.0% 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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 5000000000:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 5000000000.0) (sin x) (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 5000000000.0) {
		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 <= 5000000000.0d0) 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 <= 5000000000.0) {
		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 <= 5000000000.0:
		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 <= 5000000000.0)
		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 <= 5000000000.0)
		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, 5000000000.0], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 5e9

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 97.5%

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

   if 5e9 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 84.3%

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

Alternative 3: 54.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2600000000:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+129}:\\ \;\;\;\;x + -0.16666666666666666 \cdot \left(x \cdot \frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right)}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2600000000.0)
   (sin x)
   (if (<= y 1.5e+129)
     (+ x (* -0.16666666666666666 (* x (/ (* (* x y) (* x y)) (* y y)))))
     (/ (* x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2600000000.0) {
		tmp = sin(x);
	} else if (y <= 1.5e+129) {
		tmp = x + (-0.16666666666666666 * (x * (((x * y) * (x * y)) / (y * y))));
	} else {
		tmp = (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 <= 2600000000.0d0) then
        tmp = sin(x)
    else if (y <= 1.5d+129) then
        tmp = x + ((-0.16666666666666666d0) * (x * (((x * y) * (x * y)) / (y * y))))
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2600000000.0) {
		tmp = Math.sin(x);
	} else if (y <= 1.5e+129) {
		tmp = x + (-0.16666666666666666 * (x * (((x * y) * (x * y)) / (y * y))));
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2600000000.0:
		tmp = math.sin(x)
	elif y <= 1.5e+129:
		tmp = x + (-0.16666666666666666 * (x * (((x * y) * (x * y)) / (y * y))))
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2600000000.0)
		tmp = sin(x);
	elseif (y <= 1.5e+129)
		tmp = Float64(x + Float64(-0.16666666666666666 * Float64(x * Float64(Float64(Float64(x * y) * Float64(x * y)) / Float64(y * y)))));
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2600000000.0)
		tmp = sin(x);
	elseif (y <= 1.5e+129)
		tmp = x + (-0.16666666666666666 * (x * (((x * y) * (x * y)) / (y * y))));
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2600000000.0], N[Sin[x], $MachinePrecision], If[LessEqual[y, 1.5e+129], N[(x + N[(-0.16666666666666666 * N[(x * N[(N[(N[(x * y), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.6e9

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 58.3%

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

   if 2.6e9 < y < 1.50000000000000015e129

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 2.7%

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

   4. Taylor expanded in x around 0 15.7%

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

      1. distribute-rgt-in 15.7%

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

      2. *-lft-identity 15.7%

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

      3. associate-*l* 15.7%

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

      4. pow-plus 15.7%

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

      5. metadata-eval 15.7%

         \[\leadsto x + -0.16666666666666666 \cdot {x}^{\color{blue}{3}} \]

   6. Simplified 15.7%

      \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]
   7. Step-by-step derivation

      1. unpow3 15.7%

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

      2. pow2 15.7%

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

   8. Applied egg-rr 15.7%

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

      1. *-un-lft-identity 15.7%

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

      2. *-commutative 15.7%

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

      3. *-inverses 15.7%

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

      4. associate-/l* 15.7%

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

      5. pow2 15.7%

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

      6. frac-2neg 15.7%

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

      7. frac-times 20.1%

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

      8. *-commutative 20.1%

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

      9. distribute-rgt-neg-in 20.1%

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

   10. Applied egg-rr 20.1%

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

   if 1.50000000000000015e129 < y

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 2.7%

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

   8. Taylor expanded in x around 0 23.5%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2600000000:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+129}:\\ \;\;\;\;x + -0.16666666666666666 \cdot \left(x \cdot \frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right)}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 33.5% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+69}:\\ \;\;\;\;x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.5e+69)
   (+ x (* -0.16666666666666666 (* x (* x x))))
   (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e+69) {
		tmp = x + (-0.16666666666666666 * (x * (x * x)));
	} else {
		tmp = (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 <= 8.5d+69) then
        tmp = x + ((-0.16666666666666666d0) * (x * (x * x)))
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.5e+69) {
		tmp = x + (-0.16666666666666666 * (x * (x * x)));
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.5e+69:
		tmp = x + (-0.16666666666666666 * (x * (x * x)))
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.5e+69)
		tmp = Float64(x + Float64(-0.16666666666666666 * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.5e+69)
		tmp = x + (-0.16666666666666666 * (x * (x * x)));
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e+69], N[(x + N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5000000000000002e69

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 56.4%

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

   4. Taylor expanded in x around 0 33.9%

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

      1. distribute-rgt-in 33.9%

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

      2. *-lft-identity 33.9%

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

      3. associate-*l* 33.9%

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

      4. pow-plus 33.9%

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

      5. metadata-eval 33.9%

         \[\leadsto x + -0.16666666666666666 \cdot {x}^{\color{blue}{3}} \]

   6. Simplified 33.9%

      \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]
   7. Step-by-step derivation

      1. unpow3 33.9%

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

      2. pow2 33.9%

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

   8. Applied egg-rr 33.9%

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

      1. unpow2 33.9%

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

   10. Applied egg-rr 33.9%

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

   if 8.5000000000000002e69 < y

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 2.7%

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

   8. Taylor expanded in x around 0 21.4%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+69}:\\ \;\;\;\;x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 28.9% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+69}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.6e+69) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.6e+69) {
		tmp = x;
	} else {
		tmp = (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 <= 3.6d+69) then
        tmp = x
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.6e+69) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.6e+69:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.6e+69)
		tmp = x;
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.6e+69)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.6e+69], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.6000000000000003e69

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 63.3%

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

   4. Taylor expanded in y around 0 28.8%

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

   if 3.6000000000000003e69 < y

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 2.7%

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

   8. Taylor expanded in x around 0 21.4%

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

Alternative 6: 26.2% 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. Add Preprocessing

3. Taylor expanded in x around 0 68.3%

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

4. Taylor expanded in y around 0 23.7%

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

Reproduce

herbie shell --seed 2024160 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))