Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E

Percentage Accurate: 99.9% → 99.9%

Time: 5.5s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

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

Java

public static double code(double x, double y) {
	return (1.0 - x) + (y * Math.sqrt(x));
}

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

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

Java

public static double code(double x, double y) {
	return (1.0 - x) + (y * Math.sqrt(x));
}

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

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

Java

public static double code(double x, double y) {
	return (1.0 - x) + (y * Math.sqrt(x));
}

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(1 - x\right) + y \cdot \sqrt{x} \]

2. Final simplification 99.9%

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

Alternative 2: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+53} \lor \neg \left(y \leq 2.6 \cdot 10^{+22}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.7e+53) (not (<= y 2.6e+22)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.7e+53) || !(y <= 2.6e+22)) {
		tmp = 1.0 + (y * sqrt(x));
	} else {
		tmp = 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 <= (-3.7d+53)) .or. (.not. (y <= 2.6d+22))) then
        tmp = 1.0d0 + (y * sqrt(x))
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.7e+53) || !(y <= 2.6e+22)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.7e+53) or not (y <= 2.6e+22):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.7e+53) || !(y <= 2.6e+22))
		tmp = Float64(1.0 + Float64(y * sqrt(x)));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.7e+53) || ~((y <= 2.6e+22)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.7e+53], N[Not[LessEqual[y, 2.6e+22]], $MachinePrecision]], N[(1.0 + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7e53 or 2.6e22 < y

   1. Initial program 99.7%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Step-by-step derivation

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. add-sqr-sqrt 99.4%

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

      4. associate-*l* 99.3%

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

      5. fma-def 99.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{x}}, \sqrt{\sqrt{x}} \cdot y, 1 - x\right)} \]

      6. pow1/2 99.3%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{{x}^{0.5}}}, \sqrt{\sqrt{x}} \cdot y, 1 - x\right) \]

      7. sqrt-pow1 99.4%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}, \sqrt{\sqrt{x}} \cdot y, 1 - x\right) \]

      8. metadata-eval 99.4%

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{0.25}}, \sqrt{\sqrt{x}} \cdot y, 1 - x\right) \]

      9. pow1/2 99.4%

         \[\leadsto \mathsf{fma}\left({x}^{0.25}, \sqrt{\color{blue}{{x}^{0.5}}} \cdot y, 1 - x\right) \]

      10. sqrt-pow1 99.3%

          \[\leadsto \mathsf{fma}\left({x}^{0.25}, \color{blue}{{x}^{\left(\frac{0.5}{2}\right)}} \cdot y, 1 - x\right) \]

      11. metadata-eval 99.3%

          \[\leadsto \mathsf{fma}\left({x}^{0.25}, {x}^{\color{blue}{0.25}} \cdot y, 1 - x\right) \]

   3. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{0.25}, {x}^{0.25} \cdot y, 1 - x\right)} \]

   4. Taylor expanded in x around 0 92.0%

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

   if -3.7e53 < y < 2.6e22

   1. Initial program 100.0%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]

   2. Taylor expanded in y around 0 99.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+53} \lor \neg \left(y \leq 2.6 \cdot 10^{+22}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 3: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+57} \lor \neg \left(y \leq 1.52 \cdot 10^{+66}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.2e+57) (not (<= y 1.52e+66))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.2e+57) || !(y <= 1.52e+66)) {
		tmp = y * sqrt(x);
	} else {
		tmp = 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 <= (-4.2d+57)) .or. (.not. (y <= 1.52d+66))) then
        tmp = y * sqrt(x)
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.2e+57) || !(y <= 1.52e+66)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.2e+57) or not (y <= 1.52e+66):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.2e+57) || !(y <= 1.52e+66))
		tmp = Float64(y * sqrt(x));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.2e+57) || ~((y <= 1.52e+66)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.2e+57], N[Not[LessEqual[y, 1.52e+66]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.19999999999999982e57 or 1.52000000000000004e66 < y

   1. Initial program 99.7%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]

   2. Taylor expanded in y around inf 88.4%

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

   if -4.19999999999999982e57 < y < 1.52000000000000004e66

   1. Initial program 100.0%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]

   2. Taylor expanded in y around 0 97.5%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+57} \lor \neg \left(y \leq 1.52 \cdot 10^{+66}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 4: 69.7% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+127}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+132}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.65e+127)
   (* x (* y (- y)))
   (if (<= y 1.22e+132) (- 1.0 x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.65e+127) {
		tmp = x * (y * -y);
	} else if (y <= 1.22e+132) {
		tmp = 1.0 - x;
	} else {
		tmp = 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 <= (-1.65d+127)) then
        tmp = x * (y * -y)
    else if (y <= 1.22d+132) then
        tmp = 1.0d0 - x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.65e+127) {
		tmp = x * (y * -y);
	} else if (y <= 1.22e+132) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.65e+127:
		tmp = x * (y * -y)
	elif y <= 1.22e+132:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.65e+127)
		tmp = Float64(x * Float64(y * Float64(-y)));
	elseif (y <= 1.22e+132)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.65e+127)
		tmp = x * (y * -y);
	elseif (y <= 1.22e+132)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.65e+127], N[(x * N[(y * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.22e+132], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.64999999999999988e127

   1. Initial program 99.8%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Step-by-step derivation

      1. flip-+ 33.4%

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

      2. div-sub 33.4%

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

      3. pow2 33.4%

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

      4. associate--l- 33.4%

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

      5. *-commutative 33.4%

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

      6. *-commutative 33.4%

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

      7. swap-sqr 12.5%

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

      8. add-sqr-sqrt 12.6%

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

      9. associate--l- 12.6%

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

   3. Applied egg-rr 12.6%

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

      1. div-sub 12.6%

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

      2. associate--r+ 12.6%

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

   5. Simplified 12.6%

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

   6. Taylor expanded in y around inf 13.1%

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

      1. unpow2 13.1%

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

      2. associate-*r* 13.1%

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

      3. neg-mul-1 13.1%

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

   8. Simplified 13.1%

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

   9. Taylor expanded in x around 0 26.1%

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

       1. mul-1-neg 26.1%

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

       2. unpow2 26.1%

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

       3. *-commutative 26.1%

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

       4. distribute-rgt-neg-in 26.1%

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

       5. distribute-rgt-neg-in 26.1%

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

   11. Simplified 26.1%

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

   if -1.64999999999999988e127 < y < 1.22e132

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]

   2. Taylor expanded in y around 0 88.7%

      \[\leadsto \color{blue}{1 - x} \]

   if 1.22e132 < y

   1. Initial program 99.7%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Step-by-step derivation

      1. flip-+ 28.7%

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

      2. div-sub 28.7%

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

      3. pow2 28.7%

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

      4. associate--l- 28.7%

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

      5. *-commutative 28.7%

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

      6. *-commutative 28.7%

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

      7. swap-sqr 8.9%

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

      8. add-sqr-sqrt 8.9%

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

      9. associate--l- 8.9%

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

   3. Applied egg-rr 8.9%

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

      1. div-sub 8.9%

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

      2. associate--r+ 8.9%

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

   5. Simplified 8.9%

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

   6. Taylor expanded in y around inf 10.1%

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

      1. unpow2 10.1%

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

      2. associate-*r* 10.1%

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

      3. neg-mul-1 10.1%

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

   8. Simplified 10.1%

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

   9. Taylor expanded in x around inf 28.0%

      \[\leadsto \color{blue}{{y}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 28.0%

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

   11. Simplified 28.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+127}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+132}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 5: 69.8% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+127}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+141}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9e+127) (* y (* x (- y))) (if (<= y 8e+141) (- 1.0 x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9e+127) {
		tmp = y * (x * -y);
	} else if (y <= 8e+141) {
		tmp = 1.0 - x;
	} else {
		tmp = 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 <= (-9d+127)) then
        tmp = y * (x * -y)
    else if (y <= 8d+141) then
        tmp = 1.0d0 - x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9e+127) {
		tmp = y * (x * -y);
	} else if (y <= 8e+141) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9e+127:
		tmp = y * (x * -y)
	elif y <= 8e+141:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9e+127)
		tmp = Float64(y * Float64(x * Float64(-y)));
	elseif (y <= 8e+141)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9e+127)
		tmp = y * (x * -y);
	elseif (y <= 8e+141)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9e+127], N[(y * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+141], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.00000000000000068e127

   1. Initial program 99.8%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Step-by-step derivation

      1. flip-+ 33.4%

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

      2. div-sub 33.4%

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

      3. pow2 33.4%

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

      4. associate--l- 33.4%

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

      5. *-commutative 33.4%

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

      6. *-commutative 33.4%

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

      7. swap-sqr 12.5%

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

      8. add-sqr-sqrt 12.6%

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

      9. associate--l- 12.6%

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

   3. Applied egg-rr 12.6%

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

      1. div-sub 12.6%

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

      2. associate--r+ 12.6%

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

   5. Simplified 12.6%

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

   6. Taylor expanded in y around inf 13.1%

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

      1. unpow2 13.1%

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

      2. associate-*r* 13.1%

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

      3. neg-mul-1 13.1%

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

   8. Simplified 13.1%

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

   9. Taylor expanded in x around 0 26.1%

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

       1. mul-1-neg 26.1%

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

       2. unpow2 26.1%

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

       3. *-commutative 26.1%

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

       4. distribute-rgt-neg-in 26.1%

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

       5. distribute-rgt-neg-in 26.1%

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

   11. Simplified 26.1%

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

   12. Taylor expanded in x around 0 26.1%

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

       1. mul-1-neg 26.1%

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

       2. unpow2 26.1%

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

       3. associate-*l* 26.6%

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

       4. *-commutative 26.6%

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

       5. distribute-rgt-neg-in 26.6%

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

       6. distribute-rgt-neg-in 26.6%

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

   14. Simplified 26.6%

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

   if -9.00000000000000068e127 < y < 8.00000000000000014e141

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]

   2. Taylor expanded in y around 0 88.7%

      \[\leadsto \color{blue}{1 - x} \]

   if 8.00000000000000014e141 < y

   1. Initial program 99.7%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Step-by-step derivation

      1. flip-+ 28.7%

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

      2. div-sub 28.7%

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

      3. pow2 28.7%

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

      4. associate--l- 28.7%

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

      5. *-commutative 28.7%

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

      6. *-commutative 28.7%

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

      7. swap-sqr 8.9%

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

      8. add-sqr-sqrt 8.9%

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

      9. associate--l- 8.9%

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

   3. Applied egg-rr 8.9%

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

      1. div-sub 8.9%

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

      2. associate--r+ 8.9%

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

   5. Simplified 8.9%

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

   6. Taylor expanded in y around inf 10.1%

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

      1. unpow2 10.1%

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

      2. associate-*r* 10.1%

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

      3. neg-mul-1 10.1%

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

   8. Simplified 10.1%

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

   9. Taylor expanded in x around inf 28.0%

      \[\leadsto \color{blue}{{y}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 28.0%

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

   11. Simplified 28.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+127}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+141}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 6: 66.4% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+137}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.3e+137) (- 1.0 x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.3e+137) {
		tmp = 1.0 - x;
	} else {
		tmp = 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+137) then
        tmp = 1.0d0 - x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.3e+137) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.3e+137:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.3e+137)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.3e+137)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.3e+137], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.29999999999999999e137

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]

   2. Taylor expanded in y around 0 76.4%

      \[\leadsto \color{blue}{1 - x} \]

   if 2.29999999999999999e137 < y

   1. Initial program 99.7%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   2. Step-by-step derivation

      1. flip-+ 28.7%

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

      2. div-sub 28.7%

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

      3. pow2 28.7%

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

      4. associate--l- 28.7%

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

      5. *-commutative 28.7%

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

      6. *-commutative 28.7%

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

      7. swap-sqr 8.9%

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

      8. add-sqr-sqrt 8.9%

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

      9. associate--l- 8.9%

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

   3. Applied egg-rr 8.9%

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

      1. div-sub 8.9%

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

      2. associate--r+ 8.9%

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

   5. Simplified 8.9%

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

   6. Taylor expanded in y around inf 10.1%

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

      1. unpow2 10.1%

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

      2. associate-*r* 10.1%

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

      3. neg-mul-1 10.1%

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

   8. Simplified 10.1%

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

   9. Taylor expanded in x around inf 28.0%

      \[\leadsto \color{blue}{{y}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 28.0%

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

   11. Simplified 28.0%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+137}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 7: 61.7% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 0.0005) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0005) {
		tmp = 1.0;
	} 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 (x <= 0.0005d0) then
        tmp = 1.0d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.0005) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.0005:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0005)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.0005)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0005], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-4

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]

   2. Taylor expanded in x around 0 63.7%

      \[\leadsto \color{blue}{1} \]

   if 5.0000000000000001e-4 < x

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]

   2. Taylor expanded in x around inf 66.8%

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

      1. mul-1-neg 66.8%

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

   4. Simplified 66.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 8: 30.6% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

   \[\left(1 - x\right) + y \cdot \sqrt{x} \]

2. Taylor expanded in x around 0 32.1%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 32.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023200 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))