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

Percentage Accurate: 99.9% → 99.9%

Time: 5.5s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+42} \lor \neg \left(y \leq 7.6 \cdot 10^{+25}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1e+42) (not (<= y 7.6e+25))) (+ 1.0 (* y (sqrt x))) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1e+42) || !(y <= 7.6e+25)) {
		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 <= (-1d+42)) .or. (.not. (y <= 7.6d+25))) 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 <= -1e+42) || !(y <= 7.6e+25)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1e+42) or not (y <= 7.6e+25):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1e+42) || !(y <= 7.6e+25))
		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 <= -1e+42) || ~((y <= 7.6e+25)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1e+42], N[Not[LessEqual[y, 7.6e+25]], $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 < -1.00000000000000004e42 or 7.6000000000000001e25 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. add-sqr-sqrt 99.2%

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

      4. associate-*l* 99.2%

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

      5. fma-def 99.2%

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

      6. pow1/2 99.2%

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

      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 95.3%

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

   if -1.00000000000000004e42 < y < 7.6000000000000001e25

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+42} \lor \neg \left(y \leq 7.6 \cdot 10^{+25}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 3: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+73} \lor \neg \left(y \leq 5.5 \cdot 10^{+54}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.3e+73) (not (<= y 5.5e+54))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.3e+73) || !(y <= 5.5e+54)) {
		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 <= (-1.3d+73)) .or. (.not. (y <= 5.5d+54))) 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 <= -1.3e+73) || !(y <= 5.5e+54)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.3e+73) or not (y <= 5.5e+54):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.3e+73) || !(y <= 5.5e+54))
		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 <= -1.3e+73) || ~((y <= 5.5e+54)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.3e+73], N[Not[LessEqual[y, 5.5e+54]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e73 or 5.50000000000000026e54 < 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 99.1%

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

      4. associate-*l* 99.2%

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

      5. fma-def 99.2%

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

      6. pow1/2 99.2%

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

      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 y around inf 90.0%

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

   if -1.3e73 < y < 5.50000000000000026e54

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.7%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+73} \lor \neg \left(y \leq 5.5 \cdot 10^{+54}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 4: 68.6% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+133}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+156}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.2e+133)
   (* y (* x (- y)))
   (if (<= y 4.3e+156) (- 1.0 x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+133) {
		tmp = y * (x * -y);
	} else if (y <= 4.3e+156) {
		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 <= (-3.2d+133)) then
        tmp = y * (x * -y)
    else if (y <= 4.3d+156) 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 <= -3.2e+133) {
		tmp = y * (x * -y);
	} else if (y <= 4.3e+156) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.2e+133:
		tmp = y * (x * -y)
	elif y <= 4.3e+156:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.2e+133)
		tmp = Float64(y * Float64(x * Float64(-y)));
	elseif (y <= 4.3e+156)
		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 <= -3.2e+133)
		tmp = y * (x * -y);
	elseif (y <= 4.3e+156)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.2e+133], N[(y * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.3e+156], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.19999999999999997e133

   1. Initial program 99.6%

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

      1. flip-+ 30.5%

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

      2. div-sub 30.5%

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

      3. pow2 30.5%

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

      4. associate--l- 30.5%

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

      5. *-commutative 30.5%

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

      6. *-commutative 30.5%

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

      7. swap-sqr 13.5%

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

      8. add-sqr-sqrt 13.4%

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

      9. associate--l- 13.4%

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

   3. Applied egg-rr 13.4%

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

      1. div-sub 13.4%

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

      2. associate--r+ 13.4%

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

      3. *-commutative 13.4%

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

   5. Simplified 13.4%

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

   6. Taylor expanded in y around inf 14.9%

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

      1. mul-1-neg 14.9%

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

      2. unpow2 14.9%

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

      3. distribute-rgt-neg-in 14.9%

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

      4. distribute-rgt-neg-in 14.9%

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

   8. Simplified 14.9%

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

   9. Taylor expanded in x around 0 29.9%

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

       1. unpow2 29.9%

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

       2. associate-*r* 30.3%

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

       3. associate-*r* 30.3%

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

       4. neg-mul-1 30.3%

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

       5. *-commutative 30.3%

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

       6. distribute-rgt-neg-in 30.3%

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

   11. Simplified 30.3%

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

   if -3.19999999999999997e133 < y < 4.29999999999999985e156

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 84.2%

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

   if 4.29999999999999985e156 < y

   1. Initial program 99.7%

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

      1. flip-+ 33.2%

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

      2. div-sub 33.2%

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

      3. pow2 33.2%

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

      4. associate--l- 33.2%

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

      5. *-commutative 33.2%

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

      6. *-commutative 33.2%

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

      7. swap-sqr 3.6%

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

      8. add-sqr-sqrt 3.6%

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

      9. associate--l- 3.6%

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

   3. Applied egg-rr 3.6%

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

      1. div-sub 3.6%

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

      2. associate--r+ 3.6%

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

      3. *-commutative 3.6%

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

   5. Simplified 3.6%

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

   6. Taylor expanded in y around inf 3.8%

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

      1. mul-1-neg 3.8%

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

      2. unpow2 3.8%

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

      3. distribute-rgt-neg-in 3.8%

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

      4. distribute-rgt-neg-in 3.8%

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

   8. Simplified 3.8%

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

   9. Taylor expanded in x around inf 31.9%

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

       1. unpow2 31.9%

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

   11. Simplified 31.9%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+133}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+156}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 5: 65.3% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+156}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.85e+156) (- 1.0 x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.85e+156) {
		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 <= 1.85d+156) 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 <= 1.85e+156) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.85e+156:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.85e+156)
		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 <= 1.85e+156)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.85e+156], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.85000000000000001e156

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 74.1%

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

   if 1.85000000000000001e156 < y

   1. Initial program 99.7%

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

      1. flip-+ 33.2%

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

      2. div-sub 33.2%

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

      3. pow2 33.2%

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

      4. associate--l- 33.2%

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

      5. *-commutative 33.2%

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

      6. *-commutative 33.2%

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

      7. swap-sqr 3.6%

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

      8. add-sqr-sqrt 3.6%

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

      9. associate--l- 3.6%

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

   3. Applied egg-rr 3.6%

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

      1. div-sub 3.6%

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

      2. associate--r+ 3.6%

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

      3. *-commutative 3.6%

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

   5. Simplified 3.6%

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

   6. Taylor expanded in y around inf 3.8%

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

      1. mul-1-neg 3.8%

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

      2. unpow2 3.8%

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

      3. distribute-rgt-neg-in 3.8%

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

      4. distribute-rgt-neg-in 3.8%

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

   8. Simplified 3.8%

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

   9. Taylor expanded in x around inf 31.9%

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

       1. unpow2 31.9%

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

   11. Simplified 31.9%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+156}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 6: 60.6% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8400000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 8400000000.0) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8400000000.0) {
		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 <= 8400000000.0d0) 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 <= 8400000000.0) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 8400000000.0:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8400000000.0)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 8400000000.0)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8400000000.0], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 8.4e9

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 61.4%

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

   if 8.4e9 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 66.1%

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

      1. mul-1-neg 66.1%

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

   4. Simplified 66.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8400000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 7: 31.0% 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 34.0%

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

3. Final simplification 34.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))