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

Percentage Accurate: 99.9% → 99.9%

Time: 6.1s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+42} \lor \neg \left(y \leq 9.5 \cdot 10^{+59}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.05e+42) (not (<= y 9.5e+59)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.05e+42) || !(y <= 9.5e+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 <= (-2.05d+42)) .or. (.not. (y <= 9.5d+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 <= -2.05e+42) || !(y <= 9.5e+59)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.05e+42) or not (y <= 9.5e+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 <= -2.05e+42) || !(y <= 9.5e+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 <= -2.05e+42) || ~((y <= 9.5e+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, -2.05e+42], N[Not[LessEqual[y, 9.5e+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 < -2.05e42 or 9.50000000000000023e59 < 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* 99.3%

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

      5. fma-def 99.3%

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

      6. pow1/2 99.3%

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

      7. sqrt-pow1 99.4%

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

      8. metadata-eval 99.4%

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

      9. pow1/2 99.4%

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

      10. sqrt-pow1 99.3%

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

      11. metadata-eval 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in x around 0 97.0%

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

   if -2.05e42 < y < 9.50000000000000023e59

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+42} \lor \neg \left(y \leq 9.5 \cdot 10^{+59}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 3: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+110} \lor \neg \left(y \leq 3.2 \cdot 10^{+67}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.35e+110) (not (<= y 3.2e+67))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.35e+110) || !(y <= 3.2e+67)) {
		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 <= (-1.35d+110)) .or. (.not. (y <= 3.2d+67))) 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 <= -1.35e+110) || !(y <= 3.2e+67)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.35e+110) or not (y <= 3.2e+67):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.35e+110) || !(y <= 3.2e+67))
		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 <= -1.35e+110) || ~((y <= 3.2e+67)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.35e+110], N[Not[LessEqual[y, 3.2e+67]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35000000000000005e110 or 3.19999999999999983e67 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 95.6%

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

   if -1.35000000000000005e110 < y < 3.19999999999999983e67

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 94.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+110} \lor \neg \left(y \leq 3.2 \cdot 10^{+67}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 5: 66.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+109}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{1 + x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.85e+109) (- 1.0 x) (* y (/ x (/ (+ 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.85e+109) {
		tmp = 1.0 - x;
	} else {
		tmp = y * (x / ((1.0 + x) / 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+109) then
        tmp = 1.0d0 - x
    else
        tmp = y * (x / ((1.0d0 + x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.85e+109) {
		tmp = 1.0 - x;
	} else {
		tmp = y * (x / ((1.0 + x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.85e+109:
		tmp = 1.0 - x
	else:
		tmp = y * (x / ((1.0 + x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.85e+109)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(y * Float64(x / Float64(Float64(1.0 + x) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.85e+109)
		tmp = 1.0 - x;
	else
		tmp = y * (x / ((1.0 + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.85e+109], N[(1.0 - x), $MachinePrecision], N[(y * N[(x / N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8500000000000001e109

   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 1.8500000000000001e109 < y

   1. Initial program 99.7%

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

      1. flip-+ 36.5%

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

      2. div-sub 36.5%

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

      3. pow2 36.5%

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

      4. associate--l- 36.5%

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

      5. *-commutative 36.5%

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

      6. *-commutative 36.5%

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

      7. swap-sqr 17.8%

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

      8. add-sqr-sqrt 17.7%

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

      9. associate--l- 17.7%

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

   3. Applied egg-rr 17.7%

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

      1. div-sub 17.7%

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

      2. associate--r+ 17.7%

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

   5. Simplified 17.7%

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

   6. Taylor expanded in y around inf 18.4%

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

      1. unpow2 18.4%

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

      2. associate-*r* 18.4%

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

      3. neg-mul-1 18.4%

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

   8. Simplified 18.4%

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

   9. Taylor expanded in y around 0 28.5%

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

       1. associate-*r/ 28.5%

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

       2. mul-1-neg 28.5%

          \[\leadsto \frac{\color{blue}{-{y}^{2} \cdot x}}{1 - x} \]

       3. unpow2 28.5%

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

       4. *-commutative 28.5%

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

       5. distribute-lft-neg-in 28.5%

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

       6. associate-/l* 28.7%

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

   11. Simplified 28.7%

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

       1. associate-/r* 28.7%

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

       2. associate-/r/ 28.7%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 1.7%

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

       5. sqr-neg 1.7%

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

       6. sqrt-unprod 2.4%

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

       7. add-sqr-sqrt 2.4%

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

       8. sub-neg 2.4%

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

       9. add-sqr-sqrt 0.0%

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

       10. sqrt-unprod 13.0%

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

       11. sqr-neg 13.0%

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

       12. sqrt-unprod 30.8%

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

       13. add-sqr-sqrt 30.8%

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

   13. Applied egg-rr 30.8%

       \[\leadsto \color{blue}{\frac{x}{\frac{1 + x}{y}} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+109}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{1 + x}{y}}\\ \end{array} \]

Alternative 6: 66.0% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+109}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.85e+109) (- 1.0 x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.85e+109) {
		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+109) 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+109) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.85e+109:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.85e+109)
		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+109)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.85e+109], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8500000000000001e109

   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 1.8500000000000001e109 < y

   1. Initial program 99.7%

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

      1. flip-+ 36.5%

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

      2. div-sub 36.5%

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

      3. pow2 36.5%

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

      4. associate--l- 36.5%

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

      5. *-commutative 36.5%

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

      6. *-commutative 36.5%

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

      7. swap-sqr 17.8%

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

      8. add-sqr-sqrt 17.7%

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

      9. associate--l- 17.7%

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

   3. Applied egg-rr 17.7%

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

      1. div-sub 17.7%

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

      2. associate--r+ 17.7%

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

   5. Simplified 17.7%

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

   6. Taylor expanded in y around inf 18.4%

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

      1. unpow2 18.4%

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

      2. associate-*r* 18.4%

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

      3. neg-mul-1 18.4%

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

   8. Simplified 18.4%

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

   9. Taylor expanded in x around inf 30.4%

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

       1. unpow2 30.4%

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

   11. Simplified 30.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+109}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 7: 62.6% 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 99.8%

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

   2. Taylor expanded in x around 0 68.0%

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

   if 1 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 59.2%

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

      1. mul-1-neg 59.2%

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

   4. Simplified 59.2%

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

4. Final simplification 63.9%

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

Alternative 8: 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 99.8%

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

2. Taylor expanded in x around 0 37.2%

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

3. Final simplification 37.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023221 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))