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

Percentage Accurate: 99.9% → 99.9%

Time: 6.4s

Alternatives: 9

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: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+63} \lor \neg \left(y \leq 1.72 \cdot 10^{+59}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.1e+63) (not (<= y 1.72e+59)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.1e+63) || !(y <= 1.72e+59)) {
		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 <= (-3.1d+63)) .or. (.not. (y <= 1.72d+59))) 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 <= -3.1e+63) || !(y <= 1.72e+59)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.1e+63) or not (y <= 1.72e+59):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.1e+63) || !(y <= 1.72e+59))
		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 <= -3.1e+63) || ~((y <= 1.72e+59)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.1e+63], N[Not[LessEqual[y, 1.72e+59]], $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 < -3.1000000000000001e63 or 1.71999999999999996e59 < y

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      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 99.3%

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

      8. metadata-eval 99.3%

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

      9. pow1/2 99.3%

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

      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 94.1%

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

   if -3.1000000000000001e63 < y < 1.71999999999999996e59

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 95.5%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+63} \lor \neg \left(y \leq 1.72 \cdot 10^{+59}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 3: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+63} \lor \neg \left(y \leq 1.25 \cdot 10^{+79}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.5e+63) (not (<= y 1.25e+79))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.5e+63) || !(y <= 1.25e+79)) {
		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 <= (-2.5d+63)) .or. (.not. (y <= 1.25d+79))) 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 <= -2.5e+63) || !(y <= 1.25e+79)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.5e+63) or not (y <= 1.25e+79):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.5e+63) || !(y <= 1.25e+79))
		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 <= -2.5e+63) || ~((y <= 1.25e+79)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.5e+63], N[Not[LessEqual[y, 1.25e+79]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.50000000000000005e63 or 1.25e79 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 91.5%

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

   if -2.50000000000000005e63 < y < 1.25e79

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 95.5%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+63} \lor \neg \left(y \leq 1.25 \cdot 10^{+79}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 4: 69.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+141}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+135}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y \cdot y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.32e+141)
   (* y (* x (- y)))
   (if (<= y 1.25e+135) (- 1.0 x) (/ x (/ x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.32e+141) {
		tmp = y * (x * -y);
	} else if (y <= 1.25e+135) {
		tmp = 1.0 - x;
	} else {
		tmp = x / (x / (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.32d+141)) then
        tmp = y * (x * -y)
    else if (y <= 1.25d+135) then
        tmp = 1.0d0 - x
    else
        tmp = x / (x / (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.32e+141) {
		tmp = y * (x * -y);
	} else if (y <= 1.25e+135) {
		tmp = 1.0 - x;
	} else {
		tmp = x / (x / (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.32e+141:
		tmp = y * (x * -y)
	elif y <= 1.25e+135:
		tmp = 1.0 - x
	else:
		tmp = x / (x / (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.32e+141)
		tmp = Float64(y * Float64(x * Float64(-y)));
	elseif (y <= 1.25e+135)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(x / Float64(x / Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.32e+141)
		tmp = y * (x * -y);
	elseif (y <= 1.25e+135)
		tmp = 1.0 - x;
	else
		tmp = x / (x / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.32e+141], N[(y * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+135], N[(1.0 - x), $MachinePrecision], N[(x / N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3200000000000001e141

   1. Initial program 99.8%

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

      1. flip-+ 33.9%

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

      2. div-inv 33.9%

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

      3. pow2 33.9%

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

      4. *-commutative 33.9%

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

      5. *-commutative 33.9%

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

      6. swap-sqr 6.3%

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

      7. add-sqr-sqrt 6.2%

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

      8. associate--l- 6.2%

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

   3. Applied egg-rr 6.2%

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

   4. Taylor expanded in y around inf 6.5%

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

      1. mul-1-neg 6.5%

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

      2. unpow2 6.5%

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

      3. distribute-rgt-neg-out 6.5%

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

      4. associate-*l* 34.1%

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

   6. Simplified 34.1%

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

   7. Taylor expanded in y around 0 2.4%

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

      1. unpow2 2.4%

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

      2. *-commutative 2.4%

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

      3. associate-*r/ 2.4%

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

      4. neg-mul-1 2.4%

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

      5. distribute-rgt-neg-in 2.4%

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

      6. associate-/l* 2.4%

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

      7. distribute-rgt-neg-in 2.4%

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

   9. Simplified 2.4%

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

   10. Taylor expanded in x around 0 33.0%

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

       1. mul-1-neg 33.0%

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

       2. unpow2 33.0%

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

       3. associate-*r* 33.5%

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

       4. distribute-rgt-neg-in 33.5%

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

       5. *-commutative 33.5%

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

       6. distribute-lft-neg-out 33.5%

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

       7. *-commutative 33.5%

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

   12. Simplified 33.5%

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

   if -1.3200000000000001e141 < y < 1.25000000000000007e135

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 84.7%

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

   if 1.25000000000000007e135 < y

   1. Initial program 99.7%

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

      1. flip-+ 32.2%

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

      2. div-inv 32.0%

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

      3. pow2 32.0%

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

      4. *-commutative 32.0%

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

      5. *-commutative 32.0%

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

      6. swap-sqr 6.3%

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

      7. add-sqr-sqrt 6.3%

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

      8. associate--l- 6.3%

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

   3. Applied egg-rr 6.3%

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

   4. Taylor expanded in y around inf 7.4%

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

      1. mul-1-neg 7.4%

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

      2. unpow2 7.4%

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

      3. distribute-rgt-neg-out 7.4%

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

      4. associate-*l* 33.1%

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

   6. Simplified 33.1%

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

   7. Taylor expanded in y around 0 18.5%

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

      1. unpow2 18.5%

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

      2. *-commutative 18.5%

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

      3. associate-*r/ 18.5%

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

      4. neg-mul-1 18.5%

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

      5. distribute-rgt-neg-in 18.5%

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

      6. associate-/l* 18.6%

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

      7. distribute-rgt-neg-in 18.6%

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

   9. Simplified 18.6%

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

   10. Taylor expanded in x around inf 21.2%

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

       1. unpow2 21.2%

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

   12. Simplified 21.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+141}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+135}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y \cdot y}}\\ \end{array} \]

Alternative 5: 69.4% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+135}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.2e+141)
   (* x (* y (- y)))
   (if (<= y 2.6e+135) (- 1.0 x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.2e+141) {
		tmp = x * (y * -y);
	} else if (y <= 2.6e+135) {
		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 <= (-5.2d+141)) then
        tmp = x * (y * -y)
    else if (y <= 2.6d+135) 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 <= -5.2e+141) {
		tmp = x * (y * -y);
	} else if (y <= 2.6e+135) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.2e+141:
		tmp = x * (y * -y)
	elif y <= 2.6e+135:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.2e+141)
		tmp = Float64(x * Float64(y * Float64(-y)));
	elseif (y <= 2.6e+135)
		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 <= -5.2e+141)
		tmp = x * (y * -y);
	elseif (y <= 2.6e+135)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.2e+141], N[(x * N[(y * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+135], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.1999999999999999e141

   1. Initial program 99.8%

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

      1. flip-+ 33.9%

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

      2. div-inv 33.9%

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

      3. pow2 33.9%

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

      4. *-commutative 33.9%

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

      5. *-commutative 33.9%

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

      6. swap-sqr 6.3%

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

      7. add-sqr-sqrt 6.2%

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

      8. associate--l- 6.2%

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

   3. Applied egg-rr 6.2%

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

   4. Taylor expanded in y around inf 6.5%

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

      1. mul-1-neg 6.5%

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

      2. unpow2 6.5%

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

      3. distribute-rgt-neg-out 6.5%

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

      4. associate-*l* 34.1%

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

   6. Simplified 34.1%

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

   7. Taylor expanded in x around 0 33.0%

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

      1. unpow2 33.0%

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

      2. associate-*r* 33.0%

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

      3. neg-mul-1 33.0%

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

      4. *-commutative 33.0%

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

      5. distribute-rgt-neg-in 33.0%

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

   9. Simplified 33.0%

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

   if -5.1999999999999999e141 < y < 2.6e135

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 84.7%

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

   if 2.6e135 < y

   1. Initial program 99.7%

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

      1. flip-+ 32.2%

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

      2. div-inv 32.0%

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

      3. pow2 32.0%

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

      4. *-commutative 32.0%

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

      5. *-commutative 32.0%

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

      6. swap-sqr 6.3%

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

      7. add-sqr-sqrt 6.3%

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

      8. associate--l- 6.3%

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

   3. Applied egg-rr 6.3%

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

   4. Taylor expanded in y around inf 7.4%

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

      1. mul-1-neg 7.4%

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

      2. unpow2 7.4%

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

      3. distribute-rgt-neg-out 7.4%

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

      4. associate-*l* 33.1%

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

   6. Simplified 33.1%

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

   7. Taylor expanded in x around inf 21.2%

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

      1. unpow2 21.2%

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

   9. Simplified 21.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+135}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 6: 69.4% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+142}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+133}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e+142) (* y (* x (- y))) (if (<= y 5e+133) (- 1.0 x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+142) {
		tmp = y * (x * -y);
	} else if (y <= 5e+133) {
		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 <= (-3.8d+142)) then
        tmp = y * (x * -y)
    else if (y <= 5d+133) 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 <= -3.8e+142) {
		tmp = y * (x * -y);
	} else if (y <= 5e+133) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8e+142:
		tmp = y * (x * -y)
	elif y <= 5e+133:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.8e+142)
		tmp = Float64(y * Float64(x * Float64(-y)));
	elseif (y <= 5e+133)
		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 <= -3.8e+142)
		tmp = y * (x * -y);
	elseif (y <= 5e+133)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.8e+142], N[(y * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+133], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.7999999999999999e142

   1. Initial program 99.8%

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

      1. flip-+ 33.9%

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

      2. div-inv 33.9%

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

      3. pow2 33.9%

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

      4. *-commutative 33.9%

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

      5. *-commutative 33.9%

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

      6. swap-sqr 6.3%

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

      7. add-sqr-sqrt 6.2%

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

      8. associate--l- 6.2%

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

   3. Applied egg-rr 6.2%

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

   4. Taylor expanded in y around inf 6.5%

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

      1. mul-1-neg 6.5%

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

      2. unpow2 6.5%

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

      3. distribute-rgt-neg-out 6.5%

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

      4. associate-*l* 34.1%

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

   6. Simplified 34.1%

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

   7. Taylor expanded in y around 0 2.4%

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

      1. unpow2 2.4%

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

      2. *-commutative 2.4%

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

      3. associate-*r/ 2.4%

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

      4. neg-mul-1 2.4%

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

      5. distribute-rgt-neg-in 2.4%

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

      6. associate-/l* 2.4%

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

      7. distribute-rgt-neg-in 2.4%

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

   9. Simplified 2.4%

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

   10. Taylor expanded in x around 0 33.0%

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

       1. mul-1-neg 33.0%

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

       2. unpow2 33.0%

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

       3. associate-*r* 33.5%

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

       4. distribute-rgt-neg-in 33.5%

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

       5. *-commutative 33.5%

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

       6. distribute-lft-neg-out 33.5%

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

       7. *-commutative 33.5%

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

   12. Simplified 33.5%

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

   if -3.7999999999999999e142 < y < 4.99999999999999961e133

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 84.7%

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

   if 4.99999999999999961e133 < y

   1. Initial program 99.7%

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

      1. flip-+ 32.2%

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

      2. div-inv 32.0%

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

      3. pow2 32.0%

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

      4. *-commutative 32.0%

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

      5. *-commutative 32.0%

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

      6. swap-sqr 6.3%

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

      7. add-sqr-sqrt 6.3%

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

      8. associate--l- 6.3%

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

   3. Applied egg-rr 6.3%

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

   4. Taylor expanded in y around inf 7.4%

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

      1. mul-1-neg 7.4%

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

      2. unpow2 7.4%

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

      3. distribute-rgt-neg-out 7.4%

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

      4. associate-*l* 33.1%

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

   6. Simplified 33.1%

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

   7. Taylor expanded in x around inf 21.2%

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

      1. unpow2 21.2%

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

   9. Simplified 21.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+142}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+133}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 7: 65.7% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+135}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4.2e+135) (- 1.0 x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.2e+135) {
		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 <= 4.2d+135) 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 <= 4.2e+135) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.2e+135:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.2e+135)
		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 <= 4.2e+135)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.2e+135], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.20000000000000019e135

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 70.6%

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

   if 4.20000000000000019e135 < y

   1. Initial program 99.7%

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

      1. flip-+ 32.2%

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

      2. div-inv 32.0%

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

      3. pow2 32.0%

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

      4. *-commutative 32.0%

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

      5. *-commutative 32.0%

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

      6. swap-sqr 6.3%

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

      7. add-sqr-sqrt 6.3%

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

      8. associate--l- 6.3%

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

   3. Applied egg-rr 6.3%

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

   4. Taylor expanded in y around inf 7.4%

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

      1. mul-1-neg 7.4%

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

      2. unpow2 7.4%

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

      3. distribute-rgt-neg-out 7.4%

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

      4. associate-*l* 33.1%

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

   6. Simplified 33.1%

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

   7. Taylor expanded in x around inf 21.2%

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

      1. unpow2 21.2%

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

   9. Simplified 21.2%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+135}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 8: 61.8% accurate, 26.3× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 58.0%

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

   if 1 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 61.3%

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

      1. mul-1-neg 61.3%

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

   4. Simplified 61.3%

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

4. Final simplification 59.7%

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

Alternative 9: 32.3% 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 29.1%

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

3. Final simplification 29.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))