Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 8.7s

Alternatives: 7

Speedup: 1.0×

• Could not converge on a ground truth

  1. reg0 = (bf #e3.84137239771833519792546702963686103201e100)
  2. reg1 = (bf "1.825625460372941944624096702326987656644e256")

• 49.1% 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: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 2.0) (sin x) (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 2.0) {
		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 <= 2.0d0) 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 <= 2.0) {
		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 <= 2.0:
		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 <= 2.0)
		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 <= 2.0)
		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, 2.0], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.4%

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

   if 2 < (/.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 80.4%

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

Alternative 3: 54.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+23}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 4.15 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \left(1 + \frac{x \cdot y}{y \cdot \frac{1}{x}} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.6e+23)
   (sin x)
   (if (<= y 4.15e+98)
     (* x (+ 1.0 (* (/ (* x y) (* y (/ 1.0 x))) -0.16666666666666666)))
     (/ (* x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.6e+23) {
		tmp = sin(x);
	} else if (y <= 4.15e+98) {
		tmp = x * (1.0 + (((x * y) / (y * (1.0 / x))) * -0.16666666666666666));
	} 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 <= 1.6d+23) then
        tmp = sin(x)
    else if (y <= 4.15d+98) then
        tmp = x * (1.0d0 + (((x * y) / (y * (1.0d0 / x))) * (-0.16666666666666666d0)))
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.6e+23) {
		tmp = Math.sin(x);
	} else if (y <= 4.15e+98) {
		tmp = x * (1.0 + (((x * y) / (y * (1.0 / x))) * -0.16666666666666666));
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.6e+23:
		tmp = math.sin(x)
	elif y <= 4.15e+98:
		tmp = x * (1.0 + (((x * y) / (y * (1.0 / x))) * -0.16666666666666666))
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.6e+23)
		tmp = sin(x);
	elseif (y <= 4.15e+98)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(Float64(x * y) / Float64(y * Float64(1.0 / x))) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.6e+23)
		tmp = sin(x);
	elseif (y <= 4.15e+98)
		tmp = x * (1.0 + (((x * y) / (y * (1.0 / x))) * -0.16666666666666666));
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.6e+23], N[Sin[x], $MachinePrecision], If[LessEqual[y, 4.15e+98], N[(x * N[(1.0 + N[(N[(N[(x * y), $MachinePrecision] / N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.6e23

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 64.7%

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

   if 1.6e23 < y < 4.1499999999999998e98

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 2.5%

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

   4. Taylor expanded in x around 0 24.1%

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

      1. *-commutative 24.1%

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

   6. Simplified 24.1%

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

      1. *-rgt-identity 24.1%

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

      2. *-inverses 24.1%

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

      3. associate-/l* 24.1%

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

      4. pow2 24.1%

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

      5. clear-num 24.1%

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

      6. frac-times 24.1%

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

      7. *-un-lft-identity 24.1%

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

      8. *-commutative 24.1%

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

      9. *-commutative 24.1%

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

      10. associate-/r* 24.1%

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

      11. *-inverses 24.1%

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

   8. Applied egg-rr 24.1%

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

   if 4.1499999999999998e98 < 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 2.6%

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

   8. Taylor expanded in x around 0 25.5%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+23}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 4.15 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \left(1 + \frac{x \cdot y}{y \cdot \frac{1}{x}} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 35.2% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{+129}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot \frac{x \cdot \left(x \cdot y\right)}{y}\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+272}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.2e+129)
   (* x (+ 1.0 (* -0.16666666666666666 (/ (* x (* x y)) y))))
   (if (<= x 3.3e+272)
     (/ (* x y) y)
     (* x (+ 1.0 (* -0.16666666666666666 (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.2e+129) {
		tmp = x * (1.0 + (-0.16666666666666666 * ((x * (x * y)) / y)));
	} else if (x <= 3.3e+272) {
		tmp = (x * y) / y;
	} else {
		tmp = x * (1.0 + (-0.16666666666666666 * (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 (x <= 6.2d+129) then
        tmp = x * (1.0d0 + ((-0.16666666666666666d0) * ((x * (x * y)) / y)))
    else if (x <= 3.3d+272) then
        tmp = (x * y) / y
    else
        tmp = x * (1.0d0 + ((-0.16666666666666666d0) * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6.2e+129) {
		tmp = x * (1.0 + (-0.16666666666666666 * ((x * (x * y)) / y)));
	} else if (x <= 3.3e+272) {
		tmp = (x * y) / y;
	} else {
		tmp = x * (1.0 + (-0.16666666666666666 * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.2e+129:
		tmp = x * (1.0 + (-0.16666666666666666 * ((x * (x * y)) / y)))
	elif x <= 3.3e+272:
		tmp = (x * y) / y
	else:
		tmp = x * (1.0 + (-0.16666666666666666 * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.2e+129)
		tmp = Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * Float64(Float64(x * Float64(x * y)) / y))));
	elseif (x <= 3.3e+272)
		tmp = Float64(Float64(x * y) / y);
	else
		tmp = Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.2e+129)
		tmp = x * (1.0 + (-0.16666666666666666 * ((x * (x * y)) / y)));
	elseif (x <= 3.3e+272)
		tmp = (x * y) / y;
	else
		tmp = x * (1.0 + (-0.16666666666666666 * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.2e+129], N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e+272], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.1999999999999999e129

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 47.9%

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

   4. Taylor expanded in x around 0 30.5%

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

      1. *-commutative 30.5%

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

   6. Simplified 30.5%

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

      1. unpow2 30.5%

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

      2. *-un-lft-identity 30.5%

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

      3. associate-*r* 30.5%

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

      4. *-inverses 30.5%

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

      5. associate-/l* 31.8%

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

      6. associate-*l/ 32.2%

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

      7. *-commutative 32.2%

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

   8. Applied egg-rr 32.2%

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

   if 6.1999999999999999e129 < x < 3.29999999999999998e272

   1. Initial program 100.0%

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

      1. add-log-exp 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. log-prod 99.4%

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

      4. metadata-eval 99.4%

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

      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/ 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in y around 0 40.3%

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

   8. Taylor expanded in x around 0 36.4%

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

   if 3.29999999999999998e272 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 51.6%

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

   4. Taylor expanded in x around 0 50.9%

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

      1. *-commutative 50.9%

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

   6. Simplified 50.9%

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

      1. unpow2 50.9%

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

   8. Applied egg-rr 50.9%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{+129}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot \frac{x \cdot \left(x \cdot y\right)}{y}\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+272}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.4% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4e+98)
   (* x (+ 1.0 (* -0.16666666666666666 (* x x))))
   (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e+98) {
		tmp = x * (1.0 + (-0.16666666666666666 * (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 <= 4d+98) then
        tmp = x * (1.0d0 + ((-0.16666666666666666d0) * (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 <= 4e+98) {
		tmp = x * (1.0 + (-0.16666666666666666 * (x * x)));
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e+98:
		tmp = x * (1.0 + (-0.16666666666666666 * (x * x)))
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e+98)
		tmp = Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * 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 <= 4e+98)
		tmp = x * (1.0 + (-0.16666666666666666 * (x * x)));
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e+98], N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999999e98

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 59.2%

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

   4. Taylor expanded in x around 0 34.2%

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

      1. *-commutative 34.2%

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

   6. Simplified 34.2%

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

      1. unpow2 34.2%

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

   8. Applied egg-rr 34.2%

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

   if 3.99999999999999999e98 < 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 2.6%

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

   8. Taylor expanded in x around 0 25.5%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.6% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+81}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 3.7e+81) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.7e+81) {
		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 (x <= 3.7d+81) 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 (x <= 3.7e+81) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.7e+81:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.7e+81)
		tmp = x;
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.7e+81)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.7e+81], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.7000000000000001e81

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 72.2%

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

   4. Taylor expanded in y around 0 28.4%

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

   if 3.7000000000000001e81 < x

   1. Initial program 100.0%

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

      1. add-log-exp 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. log-prod 99.5%

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

      4. metadata-eval 99.5%

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

      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/ 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in y around 0 43.9%

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

   8. Taylor expanded in x around 0 21.9%

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

Alternative 7: 26.8% 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 65.9%

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

4. Taylor expanded in y around 0 23.9%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))