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

Percentage Accurate: 99.9% → 99.9%

Time: 5.7s

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: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+40} \lor \neg \left(y \leq 7.8 \cdot 10^{+40}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.5e+40) (not (<= y 7.8e+40)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.5e+40) || !(y <= 7.8e+40)) {
		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 <= (-4.5d+40)) .or. (.not. (y <= 7.8d+40))) 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 <= -4.5e+40) || !(y <= 7.8e+40)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.5e+40) or not (y <= 7.8e+40):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.5e+40) || !(y <= 7.8e+40))
		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 <= -4.5e+40) || ~((y <= 7.8e+40)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.5e+40], N[Not[LessEqual[y, 7.8e+40]], $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 < -4.50000000000000032e40 or 7.8000000000000002e40 < 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 99.3%

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

      4. associate-*l* 99.4%

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

      5. fma-def 99.4%

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

      6. pow1/2 99.4%

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

      7. sqrt-pow1 99.5%

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

      8. metadata-eval 99.5%

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

      9. pow1/2 99.5%

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

      10. sqrt-pow1 99.4%

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

      11. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in x around 0 91.4%

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

   if -4.50000000000000032e40 < y < 7.8000000000000002e40

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+40} \lor \neg \left(y \leq 7.8 \cdot 10^{+40}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 3: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+49} \lor \neg \left(y \leq 8 \cdot 10^{+80}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.5e+49) (not (<= y 8e+80))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e+49) || !(y <= 8e+80)) {
		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 <= (-6.5d+49)) .or. (.not. (y <= 8d+80))) 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 <= -6.5e+49) || !(y <= 8e+80)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.5e+49) or not (y <= 8e+80):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.5e+49) || !(y <= 8e+80))
		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 <= -6.5e+49) || ~((y <= 8e+80)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.5e+49], N[Not[LessEqual[y, 8e+80]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5000000000000005e49 or 8e80 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in y around inf 86.6%

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

   if -6.5000000000000005e49 < y < 8e80

   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 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+49} \lor \neg \left(y \leq 8 \cdot 10^{+80}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 4: 69.5% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+128}:\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+142}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5e+128)
   (* (- y) (* x y))
   (if (<= y 4.7e+142) (- 1.0 x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+128) {
		tmp = -y * (x * y);
	} else if (y <= 4.7e+142) {
		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 <= (-9.5d+128)) then
        tmp = -y * (x * y)
    else if (y <= 4.7d+142) 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 <= -9.5e+128) {
		tmp = -y * (x * y);
	} else if (y <= 4.7e+142) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5e+128:
		tmp = -y * (x * y)
	elif y <= 4.7e+142:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5e+128)
		tmp = Float64(Float64(-y) * Float64(x * y));
	elseif (y <= 4.7e+142)
		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 <= -9.5e+128)
		tmp = -y * (x * y);
	elseif (y <= 4.7e+142)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5e+128], N[((-y) * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.7e+142], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.50000000000000014e128

   1. Initial program 99.9%

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

      1. flip-+ 23.7%

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

      2. div-sub 23.7%

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

      3. pow2 23.7%

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

      4. associate--l- 23.7%

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

      5. *-commutative 23.7%

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

      6. *-commutative 23.7%

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

      7. swap-sqr 10.0%

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

      8. add-sqr-sqrt 10.0%

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

      9. associate--l- 10.0%

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

   3. Applied egg-rr 10.0%

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

      1. div-sub 10.0%

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

      2. associate--r+ 10.0%

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

   5. Simplified 10.0%

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

   6. Taylor expanded in y around inf 11.0%

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

      1. unpow2 11.0%

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

      2. associate-*r* 11.0%

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

      3. neg-mul-1 11.0%

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

   8. Simplified 11.0%

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

   9. Taylor expanded in x around 0 33.1%

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

       1. mul-1-neg 33.1%

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

       2. unpow2 33.1%

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

       3. associate-*r* 33.3%

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

       4. distribute-lft-neg-in 33.3%

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

   11. Simplified 33.3%

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

   if -9.50000000000000014e128 < y < 4.7e142

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 80.7%

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

   if 4.7e142 < y

   1. Initial program 99.6%

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

      1. flip-+ 26.7%

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

      2. div-sub 26.7%

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

      3. pow2 26.7%

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

      4. associate--l- 26.7%

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

      5. *-commutative 26.7%

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

      6. *-commutative 26.7%

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

      7. swap-sqr 7.1%

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

      8. add-sqr-sqrt 7.1%

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

      9. associate--l- 7.1%

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

   3. Applied egg-rr 7.1%

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

      1. div-sub 7.1%

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

      2. associate--r+ 7.1%

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

   5. Simplified 7.1%

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

   6. Taylor expanded in y around inf 7.7%

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

      1. unpow2 7.7%

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

      2. associate-*r* 7.7%

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

      3. neg-mul-1 7.7%

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

   8. Simplified 7.7%

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

   9. Taylor expanded in x around inf 21.5%

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

       1. unpow2 21.5%

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

   11. Simplified 21.5%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+128}:\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+142}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 5: 69.4% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+129}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+141}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4e+129) (* y (- y)) (if (<= y 9.8e+141) (- 1.0 x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4e+129) {
		tmp = y * -y;
	} else if (y <= 9.8e+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 <= (-1.4d+129)) then
        tmp = y * -y
    else if (y <= 9.8d+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 <= -1.4e+129) {
		tmp = y * -y;
	} else if (y <= 9.8e+141) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4e+129:
		tmp = y * -y
	elif y <= 9.8e+141:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4e+129)
		tmp = Float64(y * Float64(-y));
	elseif (y <= 9.8e+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 <= -1.4e+129)
		tmp = y * -y;
	elseif (y <= 9.8e+141)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4e+129], N[(y * (-y)), $MachinePrecision], If[LessEqual[y, 9.8e+141], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.39999999999999987e129

   1. Initial program 99.9%

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

      1. flip-+ 23.7%

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

      2. div-sub 23.7%

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

      3. pow2 23.7%

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

      4. associate--l- 23.7%

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

      5. *-commutative 23.7%

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

      6. *-commutative 23.7%

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

      7. swap-sqr 10.0%

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

      8. add-sqr-sqrt 10.0%

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

      9. associate--l- 10.0%

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

   3. Applied egg-rr 10.0%

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

      1. div-sub 10.0%

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

      2. associate--r+ 10.0%

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

   5. Simplified 10.0%

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

   6. Taylor expanded in y around inf 11.0%

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

      1. unpow2 11.0%

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

      2. associate-*r* 11.0%

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

      3. neg-mul-1 11.0%

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

   8. Simplified 11.0%

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

   9. Taylor expanded in x around inf 0.2%

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

       1. neg-mul-1 0.2%

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

   11. Simplified 0.2%

       \[\leadsto \frac{\left(-y \cdot y\right) \cdot x}{\color{blue}{-x}} \]
   12. Step-by-step derivation

       1. associate-/l* 0.2%

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

       2. neg-sub0 0.2%

          \[\leadsto \frac{\color{blue}{0 - y \cdot y}}{\frac{-x}{x}} \]

       3. div-sub 0.2%

          \[\leadsto \color{blue}{\frac{0}{\frac{-x}{x}} - \frac{y \cdot y}{\frac{-x}{x}}} \]

       4. add-sqr-sqrt 0.0%

          \[\leadsto \frac{0}{\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{x}} - \frac{y \cdot y}{\frac{-x}{x}} \]

       5. sqrt-unprod 0.1%

          \[\leadsto \frac{0}{\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{x}} - \frac{y \cdot y}{\frac{-x}{x}} \]

       6. sqr-neg 0.1%

          \[\leadsto \frac{0}{\frac{\sqrt{\color{blue}{x \cdot x}}}{x}} - \frac{y \cdot y}{\frac{-x}{x}} \]

       7. sqrt-unprod 0.2%

          \[\leadsto \frac{0}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{x}} - \frac{y \cdot y}{\frac{-x}{x}} \]

       8. add-sqr-sqrt 0.2%

          \[\leadsto \frac{0}{\frac{\color{blue}{x}}{x}} - \frac{y \cdot y}{\frac{-x}{x}} \]

       9. add-sqr-sqrt 0.2%

          \[\leadsto \frac{0}{\frac{x}{x}} - \frac{\color{blue}{\sqrt{y \cdot y} \cdot \sqrt{y \cdot y}}}{\frac{-x}{x}} \]

       10. sqrt-unprod 0.2%

           \[\leadsto \frac{0}{\frac{x}{x}} - \frac{\color{blue}{\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}}}{\frac{-x}{x}} \]

       11. sqr-neg 0.2%

           \[\leadsto \frac{0}{\frac{x}{x}} - \frac{\sqrt{\color{blue}{\left(-y \cdot y\right) \cdot \left(-y \cdot y\right)}}}{\frac{-x}{x}} \]

       12. sqrt-unprod 0.0%

           \[\leadsto \frac{0}{\frac{x}{x}} - \frac{\color{blue}{\sqrt{-y \cdot y} \cdot \sqrt{-y \cdot y}}}{\frac{-x}{x}} \]

       13. add-sqr-sqrt 32.9%

           \[\leadsto \frac{0}{\frac{x}{x}} - \frac{\color{blue}{-y \cdot y}}{\frac{-x}{x}} \]

       14. div-inv 32.9%

           \[\leadsto \frac{0}{\frac{x}{x}} - \color{blue}{\left(-y \cdot y\right) \cdot \frac{1}{\frac{-x}{x}}} \]

       15. frac-2neg 32.9%

           \[\leadsto \frac{0}{\frac{x}{x}} - \left(-y \cdot y\right) \cdot \frac{1}{\color{blue}{\frac{-\left(-x\right)}{-x}}} \]

       16. remove-double-neg 32.9%

           \[\leadsto \frac{0}{\frac{x}{x}} - \left(-y \cdot y\right) \cdot \frac{1}{\frac{\color{blue}{x}}{-x}} \]

       17. clear-num 32.9%

           \[\leadsto \frac{0}{\frac{x}{x}} - \left(-y \cdot y\right) \cdot \color{blue}{\frac{-x}{x}} \]

   13. Applied egg-rr 32.9%

       \[\leadsto \color{blue}{\frac{0}{\frac{x}{x}} - \left(y \cdot y\right) \cdot \frac{x}{x}} \]
   14. Step-by-step derivation

       1. *-inverses 32.9%

          \[\leadsto \frac{0}{\color{blue}{1}} - \left(y \cdot y\right) \cdot \frac{x}{x} \]

       2. metadata-eval 32.9%

          \[\leadsto \color{blue}{0} - \left(y \cdot y\right) \cdot \frac{x}{x} \]

       3. neg-sub0 32.9%

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

       4. *-inverses 32.9%

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

       5. *-rgt-identity 32.9%

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

       6. distribute-rgt-neg-in 32.9%

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

   15. Simplified 32.9%

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

   if -1.39999999999999987e129 < y < 9.8000000000000002e141

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 80.7%

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

   if 9.8000000000000002e141 < y

   1. Initial program 99.6%

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

      1. flip-+ 26.7%

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

      2. div-sub 26.7%

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

      3. pow2 26.7%

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

      4. associate--l- 26.7%

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

      5. *-commutative 26.7%

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

      6. *-commutative 26.7%

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

      7. swap-sqr 7.1%

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

      8. add-sqr-sqrt 7.1%

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

      9. associate--l- 7.1%

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

   3. Applied egg-rr 7.1%

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

      1. div-sub 7.1%

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

      2. associate--r+ 7.1%

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

   5. Simplified 7.1%

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

   6. Taylor expanded in y around inf 7.7%

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

      1. unpow2 7.7%

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

      2. associate-*r* 7.7%

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

      3. neg-mul-1 7.7%

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

   8. Simplified 7.7%

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

   9. Taylor expanded in x around inf 21.5%

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

       1. unpow2 21.5%

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

   11. Simplified 21.5%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+129}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+141}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 6: 66.3% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+142}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.5e+142) (- 1.0 x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.5e+142) {
		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 <= 1.5d+142) 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 <= 1.5e+142) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.5e+142:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.5e+142)
		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 <= 1.5e+142)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.5e+142], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.49999999999999987e142

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 70.5%

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

   if 1.49999999999999987e142 < y

   1. Initial program 99.6%

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

      1. flip-+ 26.7%

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

      2. div-sub 26.7%

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

      3. pow2 26.7%

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

      4. associate--l- 26.7%

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

      5. *-commutative 26.7%

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

      6. *-commutative 26.7%

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

      7. swap-sqr 7.1%

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

      8. add-sqr-sqrt 7.1%

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

      9. associate--l- 7.1%

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

   3. Applied egg-rr 7.1%

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

      1. div-sub 7.1%

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

      2. associate--r+ 7.1%

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

   5. Simplified 7.1%

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

   6. Taylor expanded in y around inf 7.7%

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

      1. unpow2 7.7%

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

      2. associate-*r* 7.7%

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

      3. neg-mul-1 7.7%

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

   8. Simplified 7.7%

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

   9. Taylor expanded in x around inf 21.5%

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

       1. unpow2 21.5%

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

   11. Simplified 21.5%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+142}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 7: 61.2% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-12}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.65e-12) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.65e-12) {
		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 <= 1.65d-12) 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 <= 1.65e-12) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.65e-12:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.65e-12)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.65e-12)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.65e-12], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 1.65e-12

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 59.4%

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

   if 1.65e-12 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 57.9%

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

      1. mul-1-neg 57.9%

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

   4. Simplified 57.9%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-12}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 8: 31.6% 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 29.6%

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

3. Final simplification 29.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023238 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))