Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 9.5s

Alternatives: 10

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 3.234625241379351e+269
  2. y = 2.0589407775864308e+290

• 50.4% 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

\[\begin{array}{l} \\ \frac{\sin x}{\frac{y}{\sinh y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (sin x) (/ y (sinh y))))

C

double code(double x, double y) {
	return sin(x) / (y / sinh(y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) / (y / sinh(y))
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) / (y / Math.sinh(y));
}

Python

def code(x, y):
	return math.sin(x) / (y / math.sinh(y))

Julia

function code(x, y)
	return Float64(sin(x) / Float64(y / sinh(y)))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) / (y / sinh(y));
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. add-log-exp 83.2%

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

   2. *-un-lft-identity 83.2%

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

   3. log-prod 83.2%

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

   4. metadata-eval 83.2%

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

   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/ 94.0%

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

   3. associate-*l/ 88.5%

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

   4. associate-/r/ 100.0%

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

6. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\sin x}{\frac{y}{\sinh y}}} \]
7. Add Preprocessing

Alternative 2: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 1.5:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.5) (sin x) (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.5) {
		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 <= 1.5d0) 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 <= 1.5) {
		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 <= 1.5:
		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 <= 1.5)
		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 <= 1.5)
		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, 1.5], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1.5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.5%

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

   if 1.5 < (/.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 77.4%

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

Alternative 3: 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 4: 69.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0054:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{+226}:\\ \;\;\;\;x \cdot \left(\sinh y \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{y \cdot \left(\sin x \cdot y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0054)
   (sin x)
   (if (<= y 6.9e+226)
     (* x (* (sinh y) (/ 1.0 y)))
     (* (/ 1.0 y) (/ (* y (* (sin x) y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0054) {
		tmp = sin(x);
	} else if (y <= 6.9e+226) {
		tmp = x * (sinh(y) * (1.0 / y));
	} else {
		tmp = (1.0 / y) * ((y * (sin(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 <= 0.0054d0) then
        tmp = sin(x)
    else if (y <= 6.9d+226) then
        tmp = x * (sinh(y) * (1.0d0 / y))
    else
        tmp = (1.0d0 / y) * ((y * (sin(x) * y)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0054) {
		tmp = Math.sin(x);
	} else if (y <= 6.9e+226) {
		tmp = x * (Math.sinh(y) * (1.0 / y));
	} else {
		tmp = (1.0 / y) * ((y * (Math.sin(x) * y)) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0054:
		tmp = math.sin(x)
	elif y <= 6.9e+226:
		tmp = x * (math.sinh(y) * (1.0 / y))
	else:
		tmp = (1.0 / y) * ((y * (math.sin(x) * y)) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0054)
		tmp = sin(x);
	elseif (y <= 6.9e+226)
		tmp = Float64(x * Float64(sinh(y) * Float64(1.0 / y)));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(Float64(y * Float64(sin(x) * y)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0054)
		tmp = sin(x);
	elseif (y <= 6.9e+226)
		tmp = x * (sinh(y) * (1.0 / y));
	else
		tmp = (1.0 / y) * ((y * (sin(x) * y)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0054], N[Sin[x], $MachinePrecision], If[LessEqual[y, 6.9e+226], N[(x * N[(N[Sinh[y], $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(N[(y * N[(N[Sin[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.0054000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 60.0%

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

   if 0.0054000000000000003 < y < 6.89999999999999959e226

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 69.3%

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

   if 6.89999999999999959e226 < 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 2.8%

      \[\leadsto \frac{\sin x \cdot \color{blue}{y}}{y} \]
   8. Step-by-step derivation

      1. clear-num 2.8%

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

      2. associate-/r/ 2.8%

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

      3. *-commutative 2.8%

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

   9. Applied egg-rr 2.8%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \left(y \cdot \sin x\right)} \]
   10. Step-by-step derivation

       1. *-commutative 2.8%

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

       2. *-un-lft-identity 2.8%

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

       3. lft-mult-inverse 2.8%

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

       4. associate-*r* 2.8%

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

       5. associate-*l/ 2.8%

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

       6. *-un-lft-identity 2.8%

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

       7. associate-*l/ 94.5%

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

   11. Applied egg-rr 94.5%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0054:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{+226}:\\ \;\;\;\;x \cdot \left(\sinh y \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{y \cdot \left(\sin x \cdot y\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.008:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sinh y \cdot \frac{1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.008) (sin x) (* x (* (sinh y) (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.008) {
		tmp = sin(x);
	} else {
		tmp = x * (sinh(y) * (1.0 / 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 <= 0.008d0) then
        tmp = sin(x)
    else
        tmp = x * (sinh(y) * (1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.008) {
		tmp = Math.sin(x);
	} else {
		tmp = x * (Math.sinh(y) * (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.008:
		tmp = math.sin(x)
	else:
		tmp = x * (math.sinh(y) * (1.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.008)
		tmp = sin(x);
	else
		tmp = Float64(x * Float64(sinh(y) * Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.008)
		tmp = sin(x);
	else
		tmp = x * (sinh(y) * (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.008], N[Sin[x], $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0080000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 60.0%

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

   if 0.0080000000000000002 < y

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 68.6%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.008:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sinh y \cdot \frac{1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+15}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 4.85 \cdot 10^{+116}:\\ \;\;\;\;x \cdot \left(\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right)}{y \cdot y} \cdot -0.16666666666666666 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(x \cdot y\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.9e+15)
   (sin x)
   (if (<= y 4.85e+116)
     (* x (+ (* (/ (* (* x y) (* x y)) (* y y)) -0.16666666666666666) 1.0))
     (/ (/ (* y (* x y)) y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.9e+15) {
		tmp = sin(x);
	} else if (y <= 4.85e+116) {
		tmp = x * (((((x * y) * (x * y)) / (y * y)) * -0.16666666666666666) + 1.0);
	} else {
		tmp = ((y * (x * y)) / 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 <= 1.9d+15) then
        tmp = sin(x)
    else if (y <= 4.85d+116) then
        tmp = x * (((((x * y) * (x * y)) / (y * y)) * (-0.16666666666666666d0)) + 1.0d0)
    else
        tmp = ((y * (x * y)) / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.9e+15) {
		tmp = Math.sin(x);
	} else if (y <= 4.85e+116) {
		tmp = x * (((((x * y) * (x * y)) / (y * y)) * -0.16666666666666666) + 1.0);
	} else {
		tmp = ((y * (x * y)) / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.9e+15:
		tmp = math.sin(x)
	elif y <= 4.85e+116:
		tmp = x * (((((x * y) * (x * y)) / (y * y)) * -0.16666666666666666) + 1.0)
	else:
		tmp = ((y * (x * y)) / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.9e+15)
		tmp = sin(x);
	elseif (y <= 4.85e+116)
		tmp = Float64(x * Float64(Float64(Float64(Float64(Float64(x * y) * Float64(x * y)) / Float64(y * y)) * -0.16666666666666666) + 1.0));
	else
		tmp = Float64(Float64(Float64(y * Float64(x * y)) / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.9e+15)
		tmp = sin(x);
	elseif (y <= 4.85e+116)
		tmp = x * (((((x * y) * (x * y)) / (y * y)) * -0.16666666666666666) + 1.0);
	else
		tmp = ((y * (x * y)) / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.9e+15], N[Sin[x], $MachinePrecision], If[LessEqual[y, 4.85e+116], N[(x * N[(N[(N[(N[(N[(x * y), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.9e15

   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 1.9e15 < y < 4.8499999999999999e116

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 2.8%

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

   4. Taylor expanded in x around 0 33.3%

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

      1. *-commutative 33.3%

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

   6. Simplified 33.3%

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

      1. unpow2 33.3%

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

   8. Applied egg-rr 33.3%

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

      1. *-rgt-identity 33.3%

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

      2. lft-mult-inverse 33.3%

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

      3. *-commutative 33.3%

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

      4. associate-*l* 33.3%

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

      5. div-inv 33.3%

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

      6. *-rgt-identity 33.3%

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

      7. lft-mult-inverse 33.3%

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

      8. *-commutative 33.3%

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

      9. associate-*l* 33.3%

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

      10. div-inv 33.3%

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

      11. frac-2neg 33.3%

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

      12. frac-times 37.5%

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

      13. *-commutative 37.5%

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

      14. distribute-rgt-neg-in 37.5%

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

   10. Applied egg-rr 37.5%

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

   if 4.8499999999999999e116 < 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 2.8%

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

   8. Taylor expanded in x around 0 24.9%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{y} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 24.9%

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

      2. *-commutative 24.9%

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

      3. lft-mult-inverse 24.9%

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

      4. associate-*l* 24.9%

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

      5. div-inv 24.9%

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

      6. associate-*l/ 43.4%

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

      7. *-commutative 43.4%

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

   10. Applied egg-rr 43.4%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+15}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 4.85 \cdot 10^{+116}:\\ \;\;\;\;x \cdot \left(\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right)}{y \cdot y} \cdot -0.16666666666666666 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(x \cdot y\right)}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.9% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.85 \cdot 10^{+116}:\\ \;\;\;\;x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(x \cdot y\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.85e+116)
   (* x (+ (* -0.16666666666666666 (* x x)) 1.0))
   (/ (/ (* y (* x y)) y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.85e+116) {
		tmp = x * ((-0.16666666666666666 * (x * x)) + 1.0);
	} else {
		tmp = ((y * (x * y)) / 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 <= 4.85d+116) then
        tmp = x * (((-0.16666666666666666d0) * (x * x)) + 1.0d0)
    else
        tmp = ((y * (x * y)) / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.85e+116) {
		tmp = x * ((-0.16666666666666666 * (x * x)) + 1.0);
	} else {
		tmp = ((y * (x * y)) / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.85e+116:
		tmp = x * ((-0.16666666666666666 * (x * x)) + 1.0)
	else:
		tmp = ((y * (x * y)) / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.85e+116)
		tmp = Float64(x * Float64(Float64(-0.16666666666666666 * Float64(x * x)) + 1.0));
	else
		tmp = Float64(Float64(Float64(y * Float64(x * y)) / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.85e+116)
		tmp = x * ((-0.16666666666666666 * (x * x)) + 1.0);
	else
		tmp = ((y * (x * y)) / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.85e+116], N[(x * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.8499999999999999e116

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 52.6%

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

   4. Taylor expanded in x around 0 30.3%

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

      1. *-commutative 30.3%

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

   6. Simplified 30.3%

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

      1. unpow2 30.3%

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

   8. Applied egg-rr 30.3%

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

   if 4.8499999999999999e116 < 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 2.8%

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

   8. Taylor expanded in x around 0 24.9%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{y} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 24.9%

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

      2. *-commutative 24.9%

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

      3. lft-mult-inverse 24.9%

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

      4. associate-*l* 24.9%

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

      5. div-inv 24.9%

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

      6. associate-*l/ 43.4%

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

      7. *-commutative 43.4%

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

   10. Applied egg-rr 43.4%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.85 \cdot 10^{+116}:\\ \;\;\;\;x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(x \cdot y\right)}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.4% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+197}:\\ \;\;\;\;x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2e+197)
   (* x (+ (* -0.16666666666666666 (* x x)) 1.0))
   (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e+197) {
		tmp = x * ((-0.16666666666666666 * (x * x)) + 1.0);
	} 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 <= 2d+197) then
        tmp = x * (((-0.16666666666666666d0) * (x * x)) + 1.0d0)
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2e+197) {
		tmp = x * ((-0.16666666666666666 * (x * x)) + 1.0);
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e+197:
		tmp = x * ((-0.16666666666666666 * (x * x)) + 1.0)
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e+197)
		tmp = Float64(x * Float64(Float64(-0.16666666666666666 * Float64(x * x)) + 1.0));
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e+197)
		tmp = x * ((-0.16666666666666666 * (x * x)) + 1.0);
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e+197], N[(x * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9999999999999999e197

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 49.5%

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

   4. Taylor expanded in x around 0 30.2%

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

      1. *-commutative 30.2%

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

   6. Simplified 30.2%

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

      1. unpow2 30.2%

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

   8. Applied egg-rr 30.2%

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

   if 1.9999999999999999e197 < 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 2.8%

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

   8. Taylor expanded in x around 0 29.7%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+197}:\\ \;\;\;\;x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.7% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+68}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 2e+68) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2e+68) {
		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 <= 2d+68) 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 <= 2e+68) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2e+68:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2e+68)
		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 <= 2e+68)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2e+68], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999991e68

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 72.0%

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

   4. Taylor expanded in y around 0 25.1%

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

   if 1.99999999999999991e68 < x

   1. Initial program 99.9%

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

      1. add-log-exp 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. log-prod 99.6%

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

      4. metadata-eval 99.6%

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

      5. add-log-exp 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. +-lft-identity 99.9%

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

      2. associate-*r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in y around 0 44.5%

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

   8. Taylor expanded in x around 0 22.8%

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

Alternative 10: 26.9% 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 62.3%

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

4. Taylor expanded in y around 0 19.8%

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

Reproduce

herbie shell --seed 2024185 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))