Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 7

Speedup: 1.0×

• 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. Final simplification 100.0%

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

Alternative 2: 69.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.3 \cdot 10^{-7}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{\sinh y}}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.3e-7) (sin x) (/ 1.0 (/ (/ y (sinh y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.3e-7) {
		tmp = sin(x);
	} else {
		tmp = 1.0 / ((y / sinh(y)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7.3d-7) then
        tmp = sin(x)
    else
        tmp = 1.0d0 / ((y / sinh(y)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.3e-7) {
		tmp = Math.sin(x);
	} else {
		tmp = 1.0 / ((y / Math.sinh(y)) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.3e-7:
		tmp = math.sin(x)
	else:
		tmp = 1.0 / ((y / math.sinh(y)) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.3e-7)
		tmp = sin(x);
	else
		tmp = Float64(1.0 / Float64(Float64(y / sinh(y)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.3e-7)
		tmp = sin(x);
	else
		tmp = 1.0 / ((y / sinh(y)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.3e-7], N[Sin[x], $MachinePrecision], N[(1.0 / N[(N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.3e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.7%

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

   if 7.3e-7 < y

   1. Initial program 99.9%

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

      1. associate-*r/ 100.0%

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

      2. clear-num 100.0%

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

      3. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 66.5%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{y}{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 66.5%

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

      2. *-commutative 66.5%

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

      3. associate-/r* 66.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{2 \cdot y}{e^{y} - \frac{1}{e^{y}}}}{x}}} \]

      4. associate-*r/ 66.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot \frac{y}{e^{y} - \frac{1}{e^{y}}}}}{x}} \]

      5. *-commutative 66.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{e^{y} - \frac{1}{e^{y}}} \cdot 2}}{x}} \]

      6. associate-/r/ 66.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{\frac{e^{y} - \frac{1}{e^{y}}}{2}}}}{x}} \]

      7. rec-exp 66.5%

         \[\leadsto \frac{1}{\frac{\frac{y}{\frac{e^{y} - \color{blue}{e^{-y}}}{2}}}{x}} \]

      8. sinh-def 67.2%

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

   7. Simplified 67.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.3 \cdot 10^{-7}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{\sinh y}}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.3 \cdot 10^{-7}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.3e-7) (sin x) (* (sinh y) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.3e-7) {
		tmp = sin(x);
	} else {
		tmp = sinh(y) * (x / 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 <= 7.3d-7) then
        tmp = sin(x)
    else
        tmp = sinh(y) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.3e-7) {
		tmp = Math.sin(x);
	} else {
		tmp = Math.sinh(y) * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.3e-7:
		tmp = math.sin(x)
	else:
		tmp = math.sinh(y) * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.3e-7)
		tmp = sin(x);
	else
		tmp = Float64(sinh(y) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.3e-7)
		tmp = sin(x);
	else
		tmp = sinh(y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.3e-7], N[Sin[x], $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.3e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.7%

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

   if 7.3e-7 < y

   1. Initial program 99.9%

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

      1. add-log-exp 97.2%

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

      2. *-un-lft-identity 97.2%

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

      3. log-prod 97.2%

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

      4. metadata-eval 97.2%

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

      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/ 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 82.1%

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

   6. Simplified 82.1%

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

   7. Taylor expanded in x around 0 49.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.3 \cdot 10^{-7}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.3 \cdot 10^{-7}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.3e-7) (sin x) (/ (* x (sinh y)) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.3e-7) {
		tmp = sin(x);
	} else {
		tmp = (x * sinh(y)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7.3d-7) then
        tmp = sin(x)
    else
        tmp = (x * sinh(y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.3e-7) {
		tmp = Math.sin(x);
	} else {
		tmp = (x * Math.sinh(y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.3e-7:
		tmp = math.sin(x)
	else:
		tmp = (x * math.sinh(y)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.3e-7)
		tmp = sin(x);
	else
		tmp = Float64(Float64(x * sinh(y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.3e-7)
		tmp = sin(x);
	else
		tmp = (x * sinh(y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.3e-7], N[Sin[x], $MachinePrecision], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.3e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.7%

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

   if 7.3e-7 < y

   1. Initial program 99.9%

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

      1. add-log-exp 97.2%

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

      2. *-un-lft-identity 97.2%

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

      3. log-prod 97.2%

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

      4. metadata-eval 97.2%

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

      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/ 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 82.1%

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

   6. Simplified 82.1%

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

   7. Taylor expanded in x around 0 49.3%

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

      1. *-commutative 49.3%

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

      2. associate-*l/ 67.2%

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

   9. Applied egg-rr 67.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.3 \cdot 10^{-7}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{+71}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4.8e+71) (sin x) (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.8e+71) {
		tmp = sin(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 <= 4.8d+71) then
        tmp = sin(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 <= 4.8e+71) {
		tmp = Math.sin(x);
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.8e+71:
		tmp = math.sin(x)
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.8e+71)
		tmp = sin(x);
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.8e+71)
		tmp = sin(x);
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.8e+71], N[Sin[x], $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.79999999999999961e71

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 69.8%

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

   if 4.79999999999999961e71 < 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%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 73.3%

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

   6. Simplified 73.3%

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

   7. Taylor expanded in y around 0 2.6%

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

      1. associate-*r/ 2.7%

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

      2. *-commutative 2.7%

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

   9. Applied egg-rr 2.7%

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

   10. Taylor expanded in x around 0 18.9%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{+71}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.4% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 5e+34) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e+34) {
		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 <= 5d+34) 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 <= 5e+34) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5e+34:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e+34)
		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 <= 5e+34)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e+34], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999998e34

   1. Initial program 100.0%

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

      1. add-log-exp 68.3%

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

      2. *-un-lft-identity 68.3%

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

      3. log-prod 68.3%

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

      4. metadata-eval 68.3%

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

      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/ 84.5%

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

      3. *-commutative 84.5%

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

      4. associate-/l* 86.7%

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

   6. Simplified 86.7%

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

   7. Taylor expanded in y around 0 59.6%

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

   8. Taylor expanded in x around 0 35.6%

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

   if 4.9999999999999998e34 < 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%

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

      3. *-commutative 99.9%

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

      4. associate-/l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 52.4%

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

      1. associate-*r/ 52.2%

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

      2. *-commutative 52.2%

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

   9. Applied egg-rr 52.2%

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

   10. Taylor expanded in x around 0 14.0%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 26.5% 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. Step-by-step derivation

   1. add-log-exp 75.5%

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

   2. *-un-lft-identity 75.5%

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

   3. log-prod 75.5%

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

   4. metadata-eval 75.5%

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

   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/ 88.0%

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

   3. *-commutative 88.0%

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

   4. associate-/l* 89.7%

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

6. Simplified 89.7%

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

7. Taylor expanded in y around 0 57.9%

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

8. Taylor expanded in x around 0 28.1%

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

9. Final simplification 28.1%

   \[\leadsto x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))