Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 9.3s

Alternatives: 9

Speedup: 1.0×

• 50.0% 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. clear-num 100.0%

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

   2. un-div-inv 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 1:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{\sinh y}}{x}}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) / y) <= 1.0) {
		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 ((sinh(y) / y) <= 1.0d0) 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 ((Math.sinh(y) / y) <= 1.0) {
		tmp = Math.sin(x);
	} else {
		tmp = 1.0 / ((y / Math.sinh(y)) / x);
	}
	return tmp;
}

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sinh(y) / y) <= 1.0)
		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 ((sinh(y) / y) <= 1.0)
		tmp = sin(x);
	else
		tmp = 1.0 / ((y / sinh(y)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], 1.0], 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 (/.f64 (sinh.f64 y) y) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   if 1 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   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 76.7%

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

      1. associate-*r/ 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-/r* 76.0%

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

      4. associate-*r/ 76.7%

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

      5. *-commutative 76.7%

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

      6. associate-/r/ 76.7%

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

      7. rec-exp 76.8%

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

      8. sinh-def 76.8%

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

   7. Simplified 76.8%

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

4. Final simplification 87.5%

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

Alternative 3: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 1:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.0) (sin x) (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		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 <= 1.0:
		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 <= 1.0)
		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 <= 1.0)
		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, 1.0], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   if 1 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   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 76.7%

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

      1. associate-*r/ 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-/r* 76.0%

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

      4. associate-*r/ 76.7%

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

      5. *-commutative 76.7%

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

      6. associate-/r/ 76.7%

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

      7. rec-exp 76.8%

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

      8. sinh-def 76.8%

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

   7. Simplified 76.8%

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

      1. associate-/r/ 76.8%

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

      2. clear-num 76.8%

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

   9. Applied egg-rr 76.8%

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

4. Final simplification 87.5%

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

Alternative 4: 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 5: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00031:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.00031) (sin x) (* (sinh y) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.00031) {
		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 <= 0.00031d0) 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 <= 0.00031) {
		tmp = Math.sin(x);
	} else {
		tmp = Math.sinh(y) * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.00031:
		tmp = math.sin(x)
	else:
		tmp = math.sinh(y) * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.00031)
		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 <= 0.00031)
		tmp = sin(x);
	else
		tmp = sinh(y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.00031], N[Sin[x], $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.1e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 66.2%

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

   if 3.1e-4 < y

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

      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 77.2%

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

      1. associate-*r/ 77.2%

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

      2. *-commutative 77.2%

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

      3. associate-/r* 77.2%

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

      4. associate-*r/ 77.2%

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

      5. *-commutative 77.2%

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

      6. associate-/r/ 77.2%

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

      7. rec-exp 77.2%

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

      8. sinh-def 77.3%

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

   7. Simplified 77.3%

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

      1. clear-num 77.3%

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

      2. associate-/r/ 54.7%

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

   9. Applied egg-rr 54.7%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00031:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 21:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 21.0) (sin x) (* (/ 1.0 y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 21.0) {
		tmp = sin(x);
	} else {
		tmp = (1.0 / 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 <= 21.0d0) then
        tmp = sin(x)
    else
        tmp = (1.0d0 / y) * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 21.0) {
		tmp = Math.sin(x);
	} else {
		tmp = (1.0 / y) * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 21.0:
		tmp = math.sin(x)
	else:
		tmp = (1.0 / y) * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 21.0)
		tmp = sin(x);
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 21.0)
		tmp = sin(x);
	else
		tmp = (1.0 / y) * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 21.0], N[Sin[x], $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 21

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 66.0%

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

   if 21 < y

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   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 y around 0 2.8%

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

   6. Taylor expanded in x around 0 7.6%

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

      1. associate-/r/ 7.6%

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

      2. *-commutative 7.6%

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

   8. Applied egg-rr 7.6%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 21:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 29.9% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8:\\ \;\;\;\;\frac{1}{x \cdot 0.16666666666666666 + \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.8)
   (/ 1.0 (+ (* x 0.16666666666666666) (/ 1.0 x)))
   (* (/ 1.0 y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.8) {
		tmp = 1.0 / ((x * 0.16666666666666666) + (1.0 / x));
	} else {
		tmp = (1.0 / 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 <= 3.8d0) then
        tmp = 1.0d0 / ((x * 0.16666666666666666d0) + (1.0d0 / x))
    else
        tmp = (1.0d0 / y) * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.8) {
		tmp = 1.0 / ((x * 0.16666666666666666) + (1.0 / x));
	} else {
		tmp = (1.0 / y) * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.8:
		tmp = 1.0 / ((x * 0.16666666666666666) + (1.0 / x))
	else:
		tmp = (1.0 / y) * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.8)
		tmp = Float64(1.0 / Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x)));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.8)
		tmp = 1.0 / ((x * 0.16666666666666666) + (1.0 / x));
	else
		tmp = (1.0 / y) * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.8], N[(1.0 / N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7999999999999998

   1. Initial program 100.0%

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

      1. associate-*r/ 83.6%

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

      2. clear-num 83.5%

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

      3. *-commutative 83.5%

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

   4. Applied egg-rr 83.5%

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

   5. Taylor expanded in y around 0 65.8%

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

   6. Taylor expanded in x around 0 34.3%

      \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 \cdot x + \frac{1}{x}}} \]

   if 3.7999999999999998 < y

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   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 y around 0 2.8%

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

   6. Taylor expanded in x around 0 7.6%

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

      1. associate-/r/ 7.6%

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

      2. *-commutative 7.6%

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

   8. Applied egg-rr 7.6%

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

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8:\\ \;\;\;\;\frac{1}{x \cdot 0.16666666666666666 + \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.5% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+53}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1e+53) x (* (/ 1.0 y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e+53) {
		tmp = x;
	} else {
		tmp = (1.0 / 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 (x <= 1d+53) then
        tmp = x
    else
        tmp = (1.0d0 / y) * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1e+53) {
		tmp = x;
	} else {
		tmp = (1.0 / y) * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1e+53:
		tmp = x
	else:
		tmp = (1.0 / y) * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e+53)
		tmp = x;
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1e+53)
		tmp = x;
	else
		tmp = (1.0 / y) * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e+53], x, N[(N[(1.0 / y), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999999e52

   1. Initial program 100.0%

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

      1. associate-*r/ 85.6%

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

      2. clear-num 85.5%

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

      3. *-commutative 85.5%

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

   4. Applied egg-rr 85.5%

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

   5. Taylor expanded in y around 0 49.2%

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

   6. Taylor expanded in x around 0 30.3%

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

   if 9.9999999999999999e52 < x

   1. Initial program 100.0%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.8%

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

      3. *-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 42.1%

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

   6. Taylor expanded in x around 0 24.7%

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

      1. associate-/r/ 24.7%

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

      2. *-commutative 24.7%

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

   8. Applied egg-rr 24.7%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+53}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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. Step-by-step derivation

   1. associate-*r/ 88.3%

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

   2. clear-num 88.2%

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

   3. *-commutative 88.2%

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

4. Applied egg-rr 88.2%

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

5. Taylor expanded in y around 0 47.8%

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

6. Taylor expanded in x around 0 25.2%

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

7. Final simplification 25.2%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))