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

Percentage Accurate: 99.9% → 99.9%

Time: 9.3s

Alternatives: 10

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 99.8%

   \[\left(1 - x\right) + y \cdot \sqrt{x} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 95.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+54} \lor \neg \left(y \leq 1.08 \cdot 10^{+63}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.02e+54) (not (<= y 1.08e+63)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.02e+54) || !(y <= 1.08e+63)) {
		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 <= (-1.02d+54)) .or. (.not. (y <= 1.08d+63))) 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 <= -1.02e+54) || !(y <= 1.08e+63)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.02e+54) or not (y <= 1.08e+63):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.02e+54) || !(y <= 1.08e+63))
		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 <= -1.02e+54) || ~((y <= 1.08e+63)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.02e+54], N[Not[LessEqual[y, 1.08e+63]], $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 < -1.02e54 or 1.08e63 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 94.2%

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

   if -1.02e54 < y < 1.08e63

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. pow2 100.0%

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

      3. pow1/2 100.0%

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

      4. sqrt-pow1 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \left(1 - x\right) + y \cdot {\left({x}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left(1 - x\right) + y \cdot \color{blue}{{\left({x}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in y around 0 99.3%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+54} \lor \neg \left(y \leq 1.08 \cdot 10^{+63}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+76} \lor \neg \left(y \leq 7.6 \cdot 10^{+63}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.95e+76) (not (<= y 7.6e+63))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.95e+76) || !(y <= 7.6e+63)) {
		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 <= (-1.95d+76)) .or. (.not. (y <= 7.6d+63))) 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 <= -1.95e+76) || !(y <= 7.6e+63)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.95e+76) or not (y <= 7.6e+63):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.95e+76) || !(y <= 7.6e+63))
		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 <= -1.95e+76) || ~((y <= 7.6e+63)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.95e+76], N[Not[LessEqual[y, 7.6e+63]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.94999999999999995e76 or 7.6000000000000002e63 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 84.5%

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

      1. associate--l+ 84.5%

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

      2. distribute-rgt-in 84.5%

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

   5. Simplified 99.5%

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

      1. sqrt-div 99.4%

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

      2. metadata-eval 99.4%

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

      3. un-div-inv 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. frac-2neg 99.5%

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

      2. clear-num 99.4%

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

      3. frac-2neg 99.4%

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

      4. clear-num 99.4%

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

      5. pow1 99.4%

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

      6. pow1/2 99.4%

         \[\leadsto \frac{1}{\frac{1}{\frac{{x}^{1}}{\color{blue}{{x}^{0.5}}}}} \cdot y + \left(1 - x\right) \]

      7. pow-div 99.5%

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{{x}^{\left(1 - 0.5\right)}}}} \cdot y + \left(1 - x\right) \]

      8. metadata-eval 99.5%

         \[\leadsto \frac{1}{\frac{1}{{x}^{\color{blue}{0.5}}}} \cdot y + \left(1 - x\right) \]

      9. pow1/2 99.5%

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

      10. associate-*l/ 99.5%

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

      11. *-un-lft-identity 99.5%

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

      12. pow1/2 99.5%

          \[\leadsto \frac{y}{\frac{1}{\color{blue}{{x}^{0.5}}}} + \left(1 - x\right) \]

      13. pow-flip 99.5%

          \[\leadsto \frac{y}{\color{blue}{{x}^{\left(-0.5\right)}}} + \left(1 - x\right) \]

      14. metadata-eval 99.5%

          \[\leadsto \frac{y}{{x}^{\color{blue}{-0.5}}} + \left(1 - x\right) \]

   9. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{y}{{x}^{-0.5}}} + \left(1 - x\right) \]

   10. Taylor expanded in y around inf 90.7%

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

   if -1.94999999999999995e76 < y < 7.6000000000000002e63

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. pow2 100.0%

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

      3. pow1/2 100.0%

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

      4. sqrt-pow1 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \left(1 - x\right) + y \cdot {\left({x}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left(1 - x\right) + y \cdot \color{blue}{{\left({x}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in y around 0 97.4%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+76} \lor \neg \left(y \leq 7.6 \cdot 10^{+63}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + \frac{y}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.0) (+ 1.0 (/ y (pow x -0.5))) (- (* y (sqrt x)) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + (y / pow(x, -0.5));
	} else {
		tmp = (y * sqrt(x)) - 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 + (y / (x ** (-0.5d0)))
    else
        tmp = (y * sqrt(x)) - 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 + (y / Math.pow(x, -0.5));
	} else {
		tmp = (y * Math.sqrt(x)) - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 + (y / math.pow(x, -0.5))
	else:
		tmp = (y * math.sqrt(x)) - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 + Float64(y / (x ^ -0.5)));
	else
		tmp = Float64(Float64(y * sqrt(x)) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 + (y / (x ^ -0.5));
	else
		tmp = (y * sqrt(x)) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.0], N[(1.0 + N[(y / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

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. Add Preprocessing

   3. Taylor expanded in x around inf 88.6%

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

      1. associate--l+ 88.6%

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

      2. distribute-rgt-in 88.6%

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

   5. Simplified 99.7%

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

      1. sqrt-div 99.7%

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

      2. metadata-eval 99.7%

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

      3. un-div-inv 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. frac-2neg 99.7%

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

      2. clear-num 99.7%

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

      3. frac-2neg 99.7%

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

      4. clear-num 99.7%

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

      5. pow1 99.7%

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

      6. pow1/2 99.7%

         \[\leadsto \frac{1}{\frac{1}{\frac{{x}^{1}}{\color{blue}{{x}^{0.5}}}}} \cdot y + \left(1 - x\right) \]

      7. pow-div 99.7%

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{{x}^{\left(1 - 0.5\right)}}}} \cdot y + \left(1 - x\right) \]

      8. metadata-eval 99.7%

         \[\leadsto \frac{1}{\frac{1}{{x}^{\color{blue}{0.5}}}} \cdot y + \left(1 - x\right) \]

      9. pow1/2 99.7%

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

      10. associate-*l/ 99.8%

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

      11. *-un-lft-identity 99.8%

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

      12. pow1/2 99.8%

          \[\leadsto \frac{y}{\frac{1}{\color{blue}{{x}^{0.5}}}} + \left(1 - x\right) \]

      13. pow-flip 99.8%

          \[\leadsto \frac{y}{\color{blue}{{x}^{\left(-0.5\right)}}} + \left(1 - x\right) \]

      14. metadata-eval 99.8%

          \[\leadsto \frac{y}{{x}^{\color{blue}{-0.5}}} + \left(1 - x\right) \]

   9. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{y}{{x}^{-0.5}}} + \left(1 - x\right) \]

   10. Taylor expanded in x around 0 99.1%

       \[\leadsto \frac{y}{{x}^{-0.5}} + \color{blue}{1} \]

   if 1 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.5%

      \[\leadsto \color{blue}{-1 \cdot x} + y \cdot \sqrt{x} \]
   4. Step-by-step derivation

      1. neg-mul-1 98.5%

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

   5. Simplified 98.5%

      \[\leadsto \color{blue}{\left(-x\right)} + y \cdot \sqrt{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + \frac{y}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x)))) (if (<= x 1.0) (+ 1.0 t_0) (- t_0 x))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(x)
    if (x <= 1.0d0) then
        tmp = 1.0d0 + t_0
    else
        tmp = t_0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * Math.sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * math.sqrt(x)
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 + t_0
	else:
		tmp = t_0 - x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * sqrt(x))
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 + t_0);
	else
		tmp = Float64(t_0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * sqrt(x);
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 + t_0;
	else
		tmp = t_0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.0], N[(1.0 + t$95$0), $MachinePrecision], N[(t$95$0 - x), $MachinePrecision]]]

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. Add Preprocessing

   3. Taylor expanded in x around 0 99.0%

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

   if 1 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.5%

      \[\leadsto \color{blue}{-1 \cdot x} + y \cdot \sqrt{x} \]
   4. Step-by-step derivation

      1. neg-mul-1 98.5%

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

   5. Simplified 98.5%

      \[\leadsto \color{blue}{\left(-x\right)} + y \cdot \sqrt{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+145}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{y}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+87}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e+145)
   (/ (* y (- x)) y)
   (if (<= y 9.2e+87) (- 1.0 x) (/ (+ y (* x y)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+145) {
		tmp = (y * -x) / y;
	} else if (y <= 9.2e+87) {
		tmp = 1.0 - x;
	} else {
		tmp = (y + (x * y)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.8d+145)) then
        tmp = (y * -x) / y
    else if (y <= 9.2d+87) then
        tmp = 1.0d0 - x
    else
        tmp = (y + (x * y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+145) {
		tmp = (y * -x) / y;
	} else if (y <= 9.2e+87) {
		tmp = 1.0 - x;
	} else {
		tmp = (y + (x * y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8e+145:
		tmp = (y * -x) / y
	elif y <= 9.2e+87:
		tmp = 1.0 - x
	else:
		tmp = (y + (x * y)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.8e+145)
		tmp = Float64(Float64(y * Float64(-x)) / y);
	elseif (y <= 9.2e+87)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(y + Float64(x * y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.8e+145)
		tmp = (y * -x) / y;
	elseif (y <= 9.2e+87)
		tmp = 1.0 - x;
	else
		tmp = (y + (x * y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.8e+145], N[(N[(y * (-x)), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 9.2e+87], N[(1.0 - x), $MachinePrecision], N[(N[(y + N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.80000000000000012e145

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf 99.6%

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

      1. associate--l+ 99.6%

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

      2. div-sub 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0 3.1%

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

      1. associate-*r/ 20.6%

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

   8. Applied egg-rr 20.6%

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

   9. Taylor expanded in x around inf 21.8%

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

       1. mul-1-neg 21.8%

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

       2. *-commutative 21.8%

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

       3. distribute-rgt-neg-in 21.8%

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

   11. Simplified 21.8%

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

   if -3.80000000000000012e145 < y < 9.2000000000000007e87

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. pow2 99.9%

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

      3. pow1/2 99.9%

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

      4. sqrt-pow1 99.9%

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

      5. metadata-eval 99.9%

         \[\leadsto \left(1 - x\right) + y \cdot {\left({x}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left(1 - x\right) + y \cdot \color{blue}{{\left({x}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in y around 0 87.9%

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

   if 9.2000000000000007e87 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf 99.6%

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

      1. associate--l+ 99.6%

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

      2. div-sub 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0 5.2%

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

      1. associate-*r/ 5.1%

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

   8. Applied egg-rr 5.1%

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

      1. sub-neg 5.1%

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

      2. distribute-lft-in 5.1%

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

      3. *-rgt-identity 5.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 27.6%

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

      6. sqr-neg 27.6%

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

      7. sqrt-unprod 27.6%

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

      8. add-sqr-sqrt 27.6%

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

   10. Applied egg-rr 27.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+145}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{y}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+87}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+143}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.8e+143) (/ (* y (- x)) y) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.8e+143) {
		tmp = (y * -x) / y;
	} 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 <= (-1.8d+143)) then
        tmp = (y * -x) / y
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.8e+143) {
		tmp = (y * -x) / y;
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.8e+143:
		tmp = (y * -x) / y
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.8e+143)
		tmp = Float64(Float64(y * Float64(-x)) / y);
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.8e+143)
		tmp = (y * -x) / y;
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.8e+143], N[(N[(y * (-x)), $MachinePrecision] / y), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e143

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf 99.6%

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

      1. associate--l+ 99.6%

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

      2. div-sub 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0 3.1%

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

      1. associate-*r/ 20.6%

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

   8. Applied egg-rr 20.6%

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

   9. Taylor expanded in x around inf 21.8%

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

       1. mul-1-neg 21.8%

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

       2. *-commutative 21.8%

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

       3. distribute-rgt-neg-in 21.8%

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

   11. Simplified 21.8%

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

   if -1.8e143 < y

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 99.8%

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

      2. pow2 99.8%

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

      3. pow1/2 99.8%

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

      4. sqrt-pow1 99.8%

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

      5. metadata-eval 99.8%

         \[\leadsto \left(1 - x\right) + y \cdot {\left({x}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left(1 - x\right) + y \cdot \color{blue}{{\left({x}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in y around 0 72.7%

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

Alternative 8: 62.2% accurate, 15.3× 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. Add Preprocessing

   3. Taylor expanded in x around 0 99.0%

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

   4. Taylor expanded in y around 0 63.1%

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

   if 1 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 76.8%

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

      1. associate--l+ 76.8%

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

      2. div-sub 76.8%

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

   5. Simplified 76.8%

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

   6. Taylor expanded in y around 0 36.9%

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

   7. Taylor expanded in x around inf 58.6%

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

      1. neg-mul-1 58.6%

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

   9. Simplified 58.6%

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

Alternative 9: 63.3% accurate, 35.7× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (- 1.0 x))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 - x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. add-sqr-sqrt 99.7%

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

   2. pow2 99.7%

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

   3. pow1/2 99.7%

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

   4. sqrt-pow1 99.7%

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

   5. metadata-eval 99.7%

      \[\leadsto \left(1 - x\right) + y \cdot {\left({x}^{\color{blue}{0.25}}\right)}^{2} \]

4. Applied egg-rr 99.7%

   \[\leadsto \left(1 - x\right) + y \cdot \color{blue}{{\left({x}^{0.25}\right)}^{2}} \]

5. Taylor expanded in y around 0 62.1%

   \[\leadsto \color{blue}{1 - x} \]
6. Add Preprocessing

Alternative 10: 31.8% 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. Add Preprocessing

3. Taylor expanded in x around 0 72.2%

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

4. Taylor expanded in y around 0 34.8%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))