Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 5.9s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return x * (Math.sin(y) / y);
}

Python

def code(x, y):
	return x * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (sin(y) / y);
end

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return x * (Math.sin(y) / y);
}

Python

def code(x, y):
	return x * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (sin(y) / y);
end

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return x * (Math.sin(y) / y);
}

Python

def code(x, y):
	return x * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (sin(y) / y);
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 63.0% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -750 \lor \neg \left(y \leq 2.3 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -750.0) (not (<= y 2.3e+55)))
   (* (/ 6.0 y) (/ x y))
   (* x (+ 1.0 (* (* y y) -0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -750.0) || !(y <= 2.3e+55)) {
		tmp = (6.0 / y) * (x / y);
	} else {
		tmp = x * (1.0 + ((y * y) * -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 <= (-750.0d0)) .or. (.not. (y <= 2.3d+55))) then
        tmp = (6.0d0 / y) * (x / y)
    else
        tmp = x * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -750.0) || !(y <= 2.3e+55)) {
		tmp = (6.0 / y) * (x / y);
	} else {
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -750.0) or not (y <= 2.3e+55):
		tmp = (6.0 / y) * (x / y)
	else:
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -750.0) || !(y <= 2.3e+55))
		tmp = Float64(Float64(6.0 / y) * Float64(x / y));
	else
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -750.0) || ~((y <= 2.3e+55)))
		tmp = (6.0 / y) * (x / y);
	else
		tmp = x * (1.0 + ((y * y) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -750.0], N[Not[LessEqual[y, 2.3e+55]], $MachinePrecision]], N[(N[(6.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -750 or 2.29999999999999987e55 < y

   1. Initial program 99.7%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around 0 32.9%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 32.9%

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

   6. Simplified 32.9%

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

   7. Taylor expanded in y around inf 32.9%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 32.9%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-*r/ 32.9%

         \[\leadsto \color{blue}{\frac{6 \cdot x}{y \cdot y}} \]

      3. times-frac 32.9%

         \[\leadsto \color{blue}{\frac{6}{y} \cdot \frac{x}{y}} \]

   9. Simplified 32.9%

      \[\leadsto \color{blue}{\frac{6}{y} \cdot \frac{x}{y}} \]

   if -750 < y < 2.29999999999999987e55

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 91.1%

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

      1. unpow2 91.1%

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

   4. Simplified 91.1%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -750 \lor \neg \left(y \leq 2.3 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 3: 63.0% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -750 \lor \neg \left(y \leq 1.05 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + -0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -750.0) (not (<= y 1.05e+56)))
   (* (/ 6.0 y) (/ x y))
   (+ x (* -0.16666666666666666 (* x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -750.0) || !(y <= 1.05e+56)) {
		tmp = (6.0 / y) * (x / y);
	} 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 <= (-750.0d0)) .or. (.not. (y <= 1.05d+56))) then
        tmp = (6.0d0 / y) * (x / y)
    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 <= -750.0) || !(y <= 1.05e+56)) {
		tmp = (6.0 / y) * (x / y);
	} else {
		tmp = x + (-0.16666666666666666 * (x * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -750.0) or not (y <= 1.05e+56):
		tmp = (6.0 / y) * (x / y)
	else:
		tmp = x + (-0.16666666666666666 * (x * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -750.0) || !(y <= 1.05e+56))
		tmp = Float64(Float64(6.0 / y) * Float64(x / y));
	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 <= -750.0) || ~((y <= 1.05e+56)))
		tmp = (6.0 / y) * (x / y);
	else
		tmp = x + (-0.16666666666666666 * (x * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -750.0], N[Not[LessEqual[y, 1.05e+56]], $MachinePrecision]], N[(N[(6.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $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 < -750 or 1.05000000000000009e56 < y

   1. Initial program 99.7%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around 0 32.9%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 32.9%

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

   6. Simplified 32.9%

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

   7. Taylor expanded in y around inf 32.9%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 32.9%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-*r/ 32.9%

         \[\leadsto \color{blue}{\frac{6 \cdot x}{y \cdot y}} \]

      3. times-frac 32.9%

         \[\leadsto \color{blue}{\frac{6}{y} \cdot \frac{x}{y}} \]

   9. Simplified 32.9%

      \[\leadsto \color{blue}{\frac{6}{y} \cdot \frac{x}{y}} \]

   if -750 < y < 1.05000000000000009e56

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 91.1%

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

      1. unpow2 91.1%

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

   4. Simplified 91.1%

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

      1. distribute-rgt-in 91.1%

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

      2. *-un-lft-identity 91.1%

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

      3. +-commutative 91.1%

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

      4. associate-*l* 91.1%

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

   6. Applied egg-rr 91.1%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -750 \lor \neg \left(y \leq 1.05 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + -0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 4: 63.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \lor \neg \left(y \leq 2.4\right):\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.5) (not (<= y 2.4))) (* 6.0 (/ x (* y y))) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.5) || !(y <= 2.4)) {
		tmp = 6.0 * (x / (y * y));
	} else {
		tmp = 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 <= (-2.5d0)) .or. (.not. (y <= 2.4d0))) then
        tmp = 6.0d0 * (x / (y * y))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.5) || !(y <= 2.4)) {
		tmp = 6.0 * (x / (y * y));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.5) or not (y <= 2.4):
		tmp = 6.0 * (x / (y * y))
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.5) || !(y <= 2.4))
		tmp = Float64(6.0 * Float64(x / Float64(y * y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.5) || ~((y <= 2.4)))
		tmp = 6.0 * (x / (y * y));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.5], N[Not[LessEqual[y, 2.4]], $MachinePrecision]], N[(6.0 * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.5 or 2.39999999999999991 < y

   1. Initial program 99.7%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around 0 29.8%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 29.8%

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

   6. Simplified 29.8%

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

   7. Taylor expanded in y around inf 29.8%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 29.8%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{y \cdot y}} \]

   9. Simplified 29.8%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{y \cdot y}} \]

   if -2.5 < y < 2.39999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.2%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \lor \neg \left(y \leq 2.4\right):\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 63.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \lor \neg \left(y \leq 2.4\right):\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.5) (not (<= y 2.4))) (* (/ 6.0 y) (/ x y)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.5) || !(y <= 2.4)) {
		tmp = (6.0 / y) * (x / y);
	} else {
		tmp = 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 <= (-2.5d0)) .or. (.not. (y <= 2.4d0))) then
        tmp = (6.0d0 / y) * (x / y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.5) || !(y <= 2.4)) {
		tmp = (6.0 / y) * (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.5) or not (y <= 2.4):
		tmp = (6.0 / y) * (x / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.5) || !(y <= 2.4))
		tmp = Float64(Float64(6.0 / y) * Float64(x / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.5) || ~((y <= 2.4)))
		tmp = (6.0 / y) * (x / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.5], N[Not[LessEqual[y, 2.4]], $MachinePrecision]], N[(N[(6.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.5 or 2.39999999999999991 < y

   1. Initial program 99.7%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around 0 29.8%

      \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 29.8%

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

   6. Simplified 29.8%

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

   7. Taylor expanded in y around inf 29.8%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 29.8%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-*r/ 29.8%

         \[\leadsto \color{blue}{\frac{6 \cdot x}{y \cdot y}} \]

      3. times-frac 29.9%

         \[\leadsto \color{blue}{\frac{6}{y} \cdot \frac{x}{y}} \]

   9. Simplified 29.9%

      \[\leadsto \color{blue}{\frac{6}{y} \cdot \frac{x}{y}} \]

   if -2.5 < y < 2.39999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.2%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \lor \neg \left(y \leq 2.4\right):\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 61.3% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+114} \lor \neg \left(y \leq 10^{-22}\right):\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1e+114) (not (<= y 1e-22))) (* y (/ x y)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1e+114) || !(y <= 1e-22)) {
		tmp = y * (x / y);
	} else {
		tmp = 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 <= (-1d+114)) .or. (.not. (y <= 1d-22))) then
        tmp = y * (x / y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1e+114) || !(y <= 1e-22)) {
		tmp = y * (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1e+114) or not (y <= 1e-22):
		tmp = y * (x / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1e+114) || !(y <= 1e-22))
		tmp = Float64(y * Float64(x / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1e+114) || ~((y <= 1e-22)))
		tmp = y * (x / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1e+114], N[Not[LessEqual[y, 1e-22]], $MachinePrecision]], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1e114 or 1e-22 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in y around 0 33.1%

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

   if -1e114 < y < 1e-22

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 88.3%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+114} \lor \neg \left(y \leq 10^{-22}\right):\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 62.4% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+34} \lor \neg \left(y \leq 1.1 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5e+34) (not (<= y 1.1e-10))) (/ y (/ y x)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5e+34) || !(y <= 1.1e-10)) {
		tmp = y / (y / x);
	} else {
		tmp = 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 <= (-5d+34)) .or. (.not. (y <= 1.1d-10))) then
        tmp = y / (y / x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5e+34) || !(y <= 1.1e-10)) {
		tmp = y / (y / x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5e+34) or not (y <= 1.1e-10):
		tmp = y / (y / x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5e+34) || !(y <= 1.1e-10))
		tmp = Float64(y / Float64(y / x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5e+34) || ~((y <= 1.1e-10)))
		tmp = y / (y / x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5e+34], N[Not[LessEqual[y, 1.1e-10]], $MachinePrecision]], N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999998e34 or 1.09999999999999995e-10 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in y around 0 28.7%

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

      1. clear-num 31.2%

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

      2. un-div-inv 31.2%

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

   7. Applied egg-rr 31.2%

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

   if -4.9999999999999998e34 < y < 1.09999999999999995e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 96.0%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+34} \lor \neg \left(y \leq 1.1 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 63.2% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ x (+ 1.0 (* 0.16666666666666666 (* y y)))))

C

double code(double x, double y) {
	return x / (1.0 + (0.16666666666666666 * (y * y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (1.0d0 + (0.16666666666666666d0 * (y * y)))
end function

Java

public static double code(double x, double y) {
	return x / (1.0 + (0.16666666666666666 * (y * y)));
}

Python

def code(x, y):
	return x / (1.0 + (0.16666666666666666 * (y * y)))

Julia

function code(x, y)
	return Float64(x / Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))))
end

MATLAB

function tmp = code(x, y)
	tmp = x / (1.0 + (0.16666666666666666 * (y * y)));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in y around 0 65.7%

   \[\leadsto \frac{x}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 65.7%

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

6. Simplified 65.7%

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

7. Final simplification 65.7%

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

Alternative 9: 51.1% accurate, 105.0× 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 99.9%

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

2. Taylor expanded in y around 0 53.3%

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

3. Final simplification 53.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023182 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))