Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 13.8s

Alternatives: 10

Speedup: 1.0×

• 49.0% 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

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

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 2: 72.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 13:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 13.0)
   (sin x)
   (if (<= y 1.4e+154)
     (* x (/ (sinh y) y))
     (* 0.16666666666666666 (* (sin x) (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 13.0) {
		tmp = sin(x);
	} else if (y <= 1.4e+154) {
		tmp = x * (sinh(y) / y);
	} else {
		tmp = 0.16666666666666666 * (sin(x) * (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 13.0d0) then
        tmp = sin(x)
    else if (y <= 1.4d+154) then
        tmp = x * (sinh(y) / y)
    else
        tmp = 0.16666666666666666d0 * (sin(x) * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 13.0) {
		tmp = Math.sin(x);
	} else if (y <= 1.4e+154) {
		tmp = x * (Math.sinh(y) / y);
	} else {
		tmp = 0.16666666666666666 * (Math.sin(x) * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 13.0:
		tmp = math.sin(x)
	elif y <= 1.4e+154:
		tmp = x * (math.sinh(y) / y)
	else:
		tmp = 0.16666666666666666 * (math.sin(x) * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 13.0)
		tmp = sin(x);
	elseif (y <= 1.4e+154)
		tmp = Float64(x * Float64(sinh(y) / y));
	else
		tmp = Float64(0.16666666666666666 * Float64(sin(x) * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 13.0)
		tmp = sin(x);
	elseif (y <= 1.4e+154)
		tmp = x * (sinh(y) / y);
	else
		tmp = 0.16666666666666666 * (sin(x) * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 13.0], N[Sin[x], $MachinePrecision], If[LessEqual[y, 1.4e+154], N[(x * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(N[Sin[x], $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 13

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 64.9%

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

   if 13 < y < 1.4e154

   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/ 82.8%

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

   3. Simplified 82.8%

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

   4. Taylor expanded in x around 0 62.1%

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

      1. associate-/r/ 79.3%

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

   6. Applied egg-rr 79.3%

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

   if 1.4e154 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({y}^{2} \cdot \sin x\right)} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 96.6%

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

      5. associate-*l* 66.9%

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

   7. Simplified 66.9%

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

   8. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({y}^{2} \cdot \sin x\right)} \]
   9. Step-by-step derivation

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 13:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 3: 84.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 13:\\ \;\;\;\;\sin x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 13.0)
   (* (sin x) (+ 1.0 (* 0.16666666666666666 (* y y))))
   (if (<= y 1.4e+154)
     (* x (/ (sinh y) y))
     (* 0.16666666666666666 (* (sin x) (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 13.0) {
		tmp = sin(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 1.4e+154) {
		tmp = x * (sinh(y) / y);
	} else {
		tmp = 0.16666666666666666 * (sin(x) * (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 13.0d0) then
        tmp = sin(x) * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    else if (y <= 1.4d+154) then
        tmp = x * (sinh(y) / y)
    else
        tmp = 0.16666666666666666d0 * (sin(x) * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 13.0) {
		tmp = Math.sin(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 1.4e+154) {
		tmp = x * (Math.sinh(y) / y);
	} else {
		tmp = 0.16666666666666666 * (Math.sin(x) * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 13.0:
		tmp = math.sin(x) * (1.0 + (0.16666666666666666 * (y * y)))
	elif y <= 1.4e+154:
		tmp = x * (math.sinh(y) / y)
	else:
		tmp = 0.16666666666666666 * (math.sin(x) * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 13.0)
		tmp = Float64(sin(x) * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	elseif (y <= 1.4e+154)
		tmp = Float64(x * Float64(sinh(y) / y));
	else
		tmp = Float64(0.16666666666666666 * Float64(sin(x) * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 13.0)
		tmp = sin(x) * (1.0 + (0.16666666666666666 * (y * y)));
	elseif (y <= 1.4e+154)
		tmp = x * (sinh(y) / y);
	else
		tmp = 0.16666666666666666 * (sin(x) * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 13.0], N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+154], N[(x * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(N[Sin[x], $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 13

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 82.5%

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

      1. unpow2 82.5%

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

   4. Simplified 82.5%

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

   if 13 < y < 1.4e154

   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/ 82.8%

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

   3. Simplified 82.8%

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

   4. Taylor expanded in x around 0 62.1%

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

      1. associate-/r/ 79.3%

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

   6. Applied egg-rr 79.3%

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

   if 1.4e154 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({y}^{2} \cdot \sin x\right)} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 96.6%

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

      5. associate-*l* 66.9%

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

   7. Simplified 66.9%

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

   8. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({y}^{2} \cdot \sin x\right)} \]
   9. Step-by-step derivation

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 13:\\ \;\;\;\;\sin x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 4: 69.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 13:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 13.0) (sin x) (* x (/ (sinh y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 13.0) {
		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 <= 13.0d0) 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 <= 13.0) {
		tmp = Math.sin(x);
	} else {
		tmp = x * (Math.sinh(y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 13.0:
		tmp = math.sin(x)
	else:
		tmp = x * (math.sinh(y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 13.0)
		tmp = sin(x);
	else
		tmp = Float64(x * Float64(sinh(y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 13.0)
		tmp = sin(x);
	else
		tmp = x * (sinh(y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 13.0], N[Sin[x], $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 13

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 64.9%

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

   if 13 < 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/ 72.2%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in x around 0 57.4%

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

      1. associate-/r/ 85.2%

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

   6. Applied egg-rr 85.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 13:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \end{array} \]

Alternative 5: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+33}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e+33) (sin x) (+ x (* 0.16666666666666666 (* x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.3e+33) {
		tmp = sin(x);
	} else {
		tmp = x + (0.16666666666666666 * (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 <= 2.3d+33) then
        tmp = sin(x)
    else
        tmp = x + (0.16666666666666666d0 * (x * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.3e+33) {
		tmp = Math.sin(x);
	} else {
		tmp = x + (0.16666666666666666 * (x * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.3e+33:
		tmp = math.sin(x)
	else:
		tmp = x + (0.16666666666666666 * (x * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.3e+33)
		tmp = sin(x);
	else
		tmp = Float64(x + Float64(0.16666666666666666 * Float64(x * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.3e+33)
		tmp = sin(x);
	else
		tmp = x + (0.16666666666666666 * (x * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.3e+33], N[Sin[x], $MachinePrecision], N[(x + N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.30000000000000011e33

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 63.7%

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

   if 2.30000000000000011e33 < 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/ 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in x around 0 56.0%

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

   5. Taylor expanded in y around 0 57.4%

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

      1. unpow2 57.4%

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

   7. Simplified 57.4%

      \[\leadsto \color{blue}{x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+33}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 6: 34.6% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 13:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.16666666666666666 \cdot \left(x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 13.0) x (* y (* 0.16666666666666666 (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 13.0) {
		tmp = x;
	} else {
		tmp = y * (0.16666666666666666 * (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 <= 13.0d0) then
        tmp = x
    else
        tmp = y * (0.16666666666666666d0 * (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 13.0) {
		tmp = x;
	} else {
		tmp = y * (0.16666666666666666 * (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 13.0:
		tmp = x
	else:
		tmp = y * (0.16666666666666666 * (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 13.0)
		tmp = x;
	else
		tmp = Float64(y * Float64(0.16666666666666666 * Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 13.0)
		tmp = x;
	else
		tmp = y * (0.16666666666666666 * (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 13.0], x, N[(y * N[(0.16666666666666666 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 13

   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/ 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around 0 48.7%

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

   5. Taylor expanded in y around 0 34.6%

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

   if 13 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 48.7%

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

      1. unpow2 48.7%

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

   4. Simplified 48.7%

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

   5. Taylor expanded in y around inf 48.7%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({y}^{2} \cdot \sin x\right)} \]
   6. Step-by-step derivation

      1. unpow2 48.7%

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

      2. associate-*r* 48.7%

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

      3. *-commutative 48.7%

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

      4. associate-*r* 47.2%

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

      5. associate-*l* 33.4%

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

   7. Simplified 33.4%

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

   8. Taylor expanded in x around 0 39.5%

      \[\leadsto y \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(x \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 13:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.16666666666666666 \cdot \left(x \cdot y\right)\right)\\ \end{array} \]

Alternative 7: 37.7% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 65:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 65.0) x (* (* y y) (* x 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 65.0) {
		tmp = x;
	} else {
		tmp = (y * y) * (x * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 65.0d0) then
        tmp = x
    else
        tmp = (y * y) * (x * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 65.0) {
		tmp = x;
	} else {
		tmp = (y * y) * (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 65.0:
		tmp = x
	else:
		tmp = (y * y) * (x * 0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 65.0)
		tmp = x;
	else
		tmp = Float64(Float64(y * y) * Float64(x * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 65.0)
		tmp = x;
	else
		tmp = (y * y) * (x * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 65.0], x, N[(N[(y * y), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 65

   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/ 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around 0 48.7%

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

   5. Taylor expanded in y around 0 34.6%

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

   if 65 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 48.7%

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

      1. unpow2 48.7%

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

   4. Simplified 48.7%

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

   5. Taylor expanded in y around inf 48.7%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({y}^{2} \cdot \sin x\right)} \]
   6. Step-by-step derivation

      1. unpow2 48.7%

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

      2. associate-*r* 48.7%

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

      3. *-commutative 48.7%

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

      4. associate-*r* 47.2%

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

      5. associate-*l* 33.4%

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

   7. Simplified 33.4%

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

   8. Taylor expanded in x around 0 39.5%

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

   9. Taylor expanded in y around 0 53.3%

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

       1. associate-*r* 53.3%

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

       2. *-commutative 53.3%

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

       3. unpow2 53.3%

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

   11. Simplified 53.3%

       \[\leadsto \color{blue}{\left(x \cdot 0.16666666666666666\right) \cdot \left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 65:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]

Alternative 8: 48.1% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (* 0.16666666666666666 (* x (* y y)))))

C

double code(double x, double y) {
	return x + (0.16666666666666666 * (x * (y * y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (0.16666666666666666d0 * (x * (y * y)))
end function

Java

public static double code(double x, double y) {
	return x + (0.16666666666666666 * (x * (y * y)));
}

Python

def code(x, y):
	return x + (0.16666666666666666 * (x * (y * y)))

Julia

function code(x, y)
	return Float64(x + Float64(0.16666666666666666 * Float64(x * Float64(y * y))))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (0.16666666666666666 * (x * (y * y)));
end

Wolfram

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

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/ 85.0%

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

3. Simplified 85.0%

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

4. Taylor expanded in x around 0 50.5%

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

5. Taylor expanded in y around 0 49.7%

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

   1. unpow2 49.7%

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

7. Simplified 49.7%

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

8. Final simplification 49.7%

   \[\leadsto x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right) \]

Alternative 9: 30.2% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 200000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 200000000000.0) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 200000000000.0) {
		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 <= 200000000000.0d0) 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 <= 200000000000.0) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 200000000000.0:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 200000000000.0)
		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 <= 200000000000.0)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 200000000000.0], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e11

   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/ 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in x around 0 49.2%

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

   5. Taylor expanded in y around 0 34.3%

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

   if 2e11 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around 0 2.7%

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

   5. Taylor expanded in x around 0 21.0%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 200000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]

Alternative 10: 27.0% 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/ 85.0%

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

3. Simplified 85.0%

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

4. Taylor expanded in x around 0 50.5%

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

5. Taylor expanded in y around 0 27.9%

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

6. Final simplification 27.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))