Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 11.8s

Alternatives: 9

Speedup: 1.0×

• 48.3% 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/ 87.9%

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

   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: 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 3: 70.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\sinh y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.45e-5) (sin x) (/ x (/ y (sinh y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.45e-5) {
		tmp = sin(x);
	} else {
		tmp = x / (y / sinh(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 <= 2.45d-5) then
        tmp = sin(x)
    else
        tmp = x / (y / sinh(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.45e-5) {
		tmp = Math.sin(x);
	} else {
		tmp = x / (y / Math.sinh(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.45e-5:
		tmp = math.sin(x)
	else:
		tmp = x / (y / math.sinh(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.45e-5)
		tmp = sin(x);
	else
		tmp = Float64(x / Float64(y / sinh(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.45e-5)
		tmp = sin(x);
	else
		tmp = x / (y / sinh(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.45e-5], N[Sin[x], $MachinePrecision], N[(x / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.45e-5

   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/ 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in y around 0 69.2%

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

   if 2.45e-5 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      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 x around 0 72.5%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\sinh y}}\\ \end{array} \]

Alternative 4: 55.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 22000000000000:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 22000000000000.0) (sin x) (* -0.16666666666666666 (pow x 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 22000000000000.0) {
		tmp = sin(x);
	} else {
		tmp = -0.16666666666666666 * pow(x, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 22000000000000.0d0) then
        tmp = sin(x)
    else
        tmp = (-0.16666666666666666d0) * (x ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 22000000000000.0) {
		tmp = Math.sin(x);
	} else {
		tmp = -0.16666666666666666 * Math.pow(x, 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 22000000000000.0:
		tmp = math.sin(x)
	else:
		tmp = -0.16666666666666666 * math.pow(x, 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 22000000000000.0)
		tmp = sin(x);
	else
		tmp = Float64(-0.16666666666666666 * (x ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 22000000000000.0)
		tmp = sin(x);
	else
		tmp = -0.16666666666666666 * (x ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 22000000000000.0], N[Sin[x], $MachinePrecision], N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.2e13

   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/ 92.2%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in y around 0 67.6%

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

   if 2.2e13 < 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/ 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in y around 0 2.7%

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

      1. clear-num 2.7%

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

      2. associate-/r/ 2.7%

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

      3. clear-num 2.7%

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

   6. Applied egg-rr 2.7%

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

   7. Taylor expanded in x around 0 28.2%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \frac{{x}^{3}}{y} + \frac{x}{y}\right)} \cdot y \]

   8. Taylor expanded in x around inf 27.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 22000000000000:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \end{array} \]

Alternative 5: 54.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{+129}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4.9e+129) (sin x) (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.9e+129) {
		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 <= 4.9d+129) 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 <= 4.9e+129) {
		tmp = Math.sin(x);
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.9e+129:
		tmp = math.sin(x)
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.9e+129)
		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 <= 4.9e+129)
		tmp = sin(x);
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.9e+129], N[Sin[x], $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9e129

   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/ 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in y around 0 63.8%

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

   if 4.9e129 < 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/ 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in y around 0 2.7%

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

      1. clear-num 2.7%

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

      2. associate-/r/ 2.7%

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

      3. clear-num 2.7%

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

   6. Applied egg-rr 2.7%

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

      1. associate-*l/ 2.8%

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

   8. Applied egg-rr 2.8%

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

   9. Taylor expanded in x around 0 7.9%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{+129}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]

Alternative 6: 30.3% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 92000:\\ \;\;\;\;\frac{y}{\left(x \cdot y\right) \cdot 0.16666666666666666 + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 92000.0)
   (/ y (+ (* (* x y) 0.16666666666666666) (/ y x)))
   (* (/ 1.0 y) (/ y (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 92000.0) {
		tmp = y / (((x * y) * 0.16666666666666666) + (y / x));
	} else {
		tmp = (1.0 / y) * (y / (1.0 / 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 <= 92000.0d0) then
        tmp = y / (((x * y) * 0.16666666666666666d0) + (y / x))
    else
        tmp = (1.0d0 / y) * (y / (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 92000.0) {
		tmp = y / (((x * y) * 0.16666666666666666) + (y / x));
	} else {
		tmp = (1.0 / y) * (y / (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 92000.0:
		tmp = y / (((x * y) * 0.16666666666666666) + (y / x))
	else:
		tmp = (1.0 / y) * (y / (1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 92000.0)
		tmp = Float64(y / Float64(Float64(Float64(x * y) * 0.16666666666666666) + Float64(y / x)));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(y / Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 92000.0)
		tmp = y / (((x * y) * 0.16666666666666666) + (y / x));
	else
		tmp = (1.0 / y) * (y / (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 92000.0], N[(y / N[(N[(N[(x * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(y / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 92000

   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/ 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in y around 0 69.0%

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

   5. Taylor expanded in x around 0 38.3%

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

   if 92000 < 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/ 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in y around 0 2.7%

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

   5. Taylor expanded in x around 0 2.2%

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

      1. *-un-lft-identity 2.2%

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

      2. div-inv 2.2%

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

      3. times-frac 6.0%

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

   7. Applied egg-rr 6.0%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 92000:\\ \;\;\;\;\frac{y}{\left(x \cdot y\right) \cdot 0.16666666666666666 + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \]

Alternative 7: 30.3% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9000:\\ \;\;\;\;\frac{1}{\frac{1}{x} + x \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9000.0)
   (/ 1.0 (+ (/ 1.0 x) (* x 0.16666666666666666)))
   (* (/ 1.0 y) (/ y (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9000.0) {
		tmp = 1.0 / ((1.0 / x) + (x * 0.16666666666666666));
	} else {
		tmp = (1.0 / y) * (y / (1.0 / 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 <= 9000.0d0) then
        tmp = 1.0d0 / ((1.0d0 / x) + (x * 0.16666666666666666d0))
    else
        tmp = (1.0d0 / y) * (y / (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9000.0) {
		tmp = 1.0 / ((1.0 / x) + (x * 0.16666666666666666));
	} else {
		tmp = (1.0 / y) * (y / (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9000.0:
		tmp = 1.0 / ((1.0 / x) + (x * 0.16666666666666666))
	else:
		tmp = (1.0 / y) * (y / (1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9000.0)
		tmp = Float64(1.0 / Float64(Float64(1.0 / x) + Float64(x * 0.16666666666666666)));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(y / Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9000.0)
		tmp = 1.0 / ((1.0 / x) + (x * 0.16666666666666666));
	else
		tmp = (1.0 / y) * (y / (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9000.0], N[(1.0 / N[(N[(1.0 / x), $MachinePrecision] + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(y / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9e3

   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/ 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in y around 0 69.0%

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

   5. Taylor expanded in x around 0 38.3%

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

   6. Taylor expanded in y around 0 38.3%

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

   if 9e3 < 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/ 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in y around 0 2.7%

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

   5. Taylor expanded in x around 0 2.2%

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

      1. *-un-lft-identity 2.2%

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

      2. div-inv 2.2%

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

      3. times-frac 6.0%

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

   7. Applied egg-rr 6.0%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9000:\\ \;\;\;\;\frac{1}{\frac{1}{x} + x \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \]

Alternative 8: 29.7% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.08 \cdot 10^{+129}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.08e+129) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.08e+129) {
		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 <= 1.08d+129) 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 <= 1.08e+129) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.08e+129:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.08e+129)
		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 <= 1.08e+129)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.08e+129], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.08e129

   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/ 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in y around 0 63.6%

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

   5. Taylor expanded in x around 0 35.3%

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

   if 1.08e129 < 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/ 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in y around 0 2.7%

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

      1. clear-num 2.7%

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

      2. associate-/r/ 2.7%

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

      3. clear-num 2.7%

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

   6. Applied egg-rr 2.7%

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

      1. associate-*l/ 2.8%

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

   8. Applied egg-rr 2.8%

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

   9. Taylor expanded in x around 0 7.9%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.08 \cdot 10^{+129}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]

Alternative 9: 27.1% 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/ 90.1%

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

3. Simplified 90.1%

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

4. Taylor expanded in y around 0 56.0%

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

5. Taylor expanded in x around 0 31.2%

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

6. Final simplification 31.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023308 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))