Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 8

Speedup: 1.0×

• 49.6% 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. Step-by-step derivation

   1. associate-*r/ 89.7%

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

   2. associate-/l* 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 86.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 2.0) {
		tmp = sin(x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sinh(y) / y
    if (t_0 <= 2.0d0) then
        tmp = sin(x)
    else
        tmp = x * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(y) / y;
	double tmp;
	if (t_0 <= 2.0) {
		tmp = Math.sin(x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if t_0 <= 2.0:
		tmp = math.sin(x)
	else:
		tmp = x * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = sin(x);
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = sin(x);
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. associate-*r/ 79.2%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.7%

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

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

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in x around 0 56.6%

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

      1. associate-/r/ 79.8%

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

   6. Applied egg-rr 79.8%

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

4. Final simplification 89.2%

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

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. Final simplification 100.0%

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

Alternative 4: 53.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+53}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.1e+53) (sin x) (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e+53) {
		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 <= 2.1d+53) 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 <= 2.1e+53) {
		tmp = Math.sin(x);
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e+53:
		tmp = math.sin(x)
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e+53)
		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 <= 2.1e+53)
		tmp = sin(x);
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e+53], N[Sin[x], $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1000000000000002e53

   1. Initial program 100.0%

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

      1. associate-*r/ 86.7%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 64.3%

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

   if 2.1000000000000002e53 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 75.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in y around 0 2.6%

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

   5. Taylor expanded in x around 0 2.6%

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

      1. *-un-lft-identity 2.6%

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

      2. div-inv 2.6%

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

      3. times-frac 27.1%

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

   7. Applied egg-rr 27.1%

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

      1. frac-times 2.6%

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

      2. *-un-lft-identity 2.6%

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

      3. div-inv 2.6%

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

      4. associate-/l* 27.1%

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

   9. Applied egg-rr 27.1%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+53}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]

Alternative 5: 29.5% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 55:\\ \;\;\;\;\frac{y}{0.16666666666666666 \cdot \left(x \cdot y\right) + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 55.0)
   (/ y (+ (* 0.16666666666666666 (* x y)) (/ y x)))
   (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 55.0) {
		tmp = y / ((0.16666666666666666 * (x * y)) + (y / 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 <= 55.0d0) then
        tmp = y / ((0.16666666666666666d0 * (x * y)) + (y / 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 <= 55.0) {
		tmp = y / ((0.16666666666666666 * (x * y)) + (y / x));
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 55.0:
		tmp = y / ((0.16666666666666666 * (x * y)) + (y / x))
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 55.0)
		tmp = Float64(y / Float64(Float64(0.16666666666666666 * Float64(x * y)) + Float64(y / x)));
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 55.0)
		tmp = y / ((0.16666666666666666 * (x * y)) + (y / x));
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 55.0], N[(y / N[(N[(0.16666666666666666 * N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 55

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in y around 0 67.7%

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

   5. Taylor expanded in x around 0 33.4%

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

   if 55 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in y around 0 2.6%

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

   5. Taylor expanded in x around 0 2.7%

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

      1. *-un-lft-identity 2.7%

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

      2. div-inv 2.7%

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

      3. times-frac 23.3%

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

   7. Applied egg-rr 23.3%

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

      1. frac-times 2.7%

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

      2. *-un-lft-identity 2.7%

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

      3. div-inv 2.7%

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

      4. associate-/l* 23.3%

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

   9. Applied egg-rr 23.3%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 55:\\ \;\;\;\;\frac{y}{0.16666666666666666 \cdot \left(x \cdot y\right) + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]

Alternative 6: 29.5% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 59:\\ \;\;\;\;\frac{1}{x \cdot 0.16666666666666666 + \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 59.0)
   (/ 1.0 (+ (* x 0.16666666666666666) (/ 1.0 x)))
   (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 59.0) {
		tmp = 1.0 / ((x * 0.16666666666666666) + (1.0 / 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 <= 59.0d0) then
        tmp = 1.0d0 / ((x * 0.16666666666666666d0) + (1.0d0 / 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 <= 59.0) {
		tmp = 1.0 / ((x * 0.16666666666666666) + (1.0 / x));
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 59.0:
		tmp = 1.0 / ((x * 0.16666666666666666) + (1.0 / x))
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 59.0)
		tmp = Float64(1.0 / Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x)));
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 59.0)
		tmp = 1.0 / ((x * 0.16666666666666666) + (1.0 / x));
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 59.0], N[(1.0 / N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 59

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in y around 0 67.7%

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

   5. Taylor expanded in x around 0 33.4%

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

   6. Taylor expanded in y around 0 33.4%

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

   if 59 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in y around 0 2.6%

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

   5. Taylor expanded in x around 0 2.7%

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

      1. *-un-lft-identity 2.7%

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

      2. div-inv 2.7%

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

      3. times-frac 23.3%

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

   7. Applied egg-rr 23.3%

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

      1. frac-times 2.7%

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

      2. *-un-lft-identity 2.7%

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

      3. div-inv 2.7%

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

      4. associate-/l* 23.3%

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

   9. Applied egg-rr 23.3%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 59:\\ \;\;\;\;\frac{1}{x \cdot 0.16666666666666666 + \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]

Alternative 7: 29.3% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+41}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1e+41) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e+41) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1d+41) then
        tmp = x
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1e+41) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e+41:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e+41)
		tmp = x;
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e+41)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e+41], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000001e41

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in y around 0 64.7%

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

   5. Taylor expanded in x around 0 32.1%

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

   if 1.00000000000000001e41 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in y around 0 2.6%

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

   5. Taylor expanded in x around 0 2.6%

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

      1. *-un-lft-identity 2.6%

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

      2. div-inv 2.6%

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

      3. times-frac 26.3%

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

   7. Applied egg-rr 26.3%

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

      1. frac-times 2.6%

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

      2. *-un-lft-identity 2.6%

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

      3. div-inv 2.6%

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

      4. associate-/l* 26.3%

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

   9. Applied egg-rr 26.3%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+41}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]

Alternative 8: 26.3% 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. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. associate-/r/ 88.1%

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

3. Simplified 88.1%

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

4. Taylor expanded in y around 0 50.1%

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

5. Taylor expanded in x around 0 25.2%

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

6. Final simplification 25.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))