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

Percentage Accurate: 99.9% → 99.9%

Time: 8.0s

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: 94.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+23} \lor \neg \left(y \leq 6.6 \cdot 10^{+69}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.16e+23) (not (<= y 6.6e+69)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.16e+23) || !(y <= 6.6e+69)) {
		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 <= (-1.16d+23)) .or. (.not. (y <= 6.6d+69))) 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 <= -1.16e+23) || !(y <= 6.6e+69)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.16e+23) or not (y <= 6.6e+69):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.16e+23) || !(y <= 6.6e+69))
		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 <= -1.16e+23) || ~((y <= 6.6e+69)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.16e+23], N[Not[LessEqual[y, 6.6e+69]], $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 < -1.16e23 or 6.5999999999999997e69 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 92.5%

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

   if -1.16e23 < y < 6.5999999999999997e69

   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 98.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+23} \lor \neg \left(y \leq 6.6 \cdot 10^{+69}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+68}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+77}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{{x}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.5e+68)
   (* y (/ x (sqrt x)))
   (if (<= y 9.2e+77) (- 1.0 x) (/ y (pow x -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.5e+68) {
		tmp = y * (x / sqrt(x));
	} else if (y <= 9.2e+77) {
		tmp = 1.0 - x;
	} else {
		tmp = y / pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.5d+68)) then
        tmp = y * (x / sqrt(x))
    else if (y <= 9.2d+77) then
        tmp = 1.0d0 - x
    else
        tmp = y / (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.5e+68) {
		tmp = y * (x / Math.sqrt(x));
	} else if (y <= 9.2e+77) {
		tmp = 1.0 - x;
	} else {
		tmp = y / Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.5e+68:
		tmp = y * (x / math.sqrt(x))
	elif y <= 9.2e+77:
		tmp = 1.0 - x
	else:
		tmp = y / math.pow(x, -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.5e+68)
		tmp = Float64(y * Float64(x / sqrt(x)));
	elseif (y <= 9.2e+77)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(y / (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.5e+68)
		tmp = y * (x / sqrt(x));
	elseif (y <= 9.2e+77)
		tmp = 1.0 - x;
	else
		tmp = y / (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.5e+68], N[(y * N[(x / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.2e+77], N[(1.0 - x), $MachinePrecision], N[(y / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5000000000000001e68

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 88.3%

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

   4. Taylor expanded in x around 0 83.2%

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

      1. associate-*r* 92.5%

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

      2. sqrt-div 92.5%

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

      3. metadata-eval 92.5%

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

      4. div-inv 92.6%

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

      5. *-commutative 92.6%

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

      6. associate-*r/ 82.9%

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

      7. *-commutative 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-/l* 92.6%

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

   8. Applied egg-rr 92.6%

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

   if -1.5000000000000001e68 < y < 9.19999999999999979e77

   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 95.6%

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

   if 9.19999999999999979e77 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 76.7%

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

   4. Taylor expanded in x around 0 70.0%

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

      1. associate-*r* 86.9%

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

      2. sqrt-div 86.9%

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

      3. metadata-eval 86.9%

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

      4. div-inv 86.8%

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

      5. *-commutative 86.8%

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

      6. associate-*r/ 73.3%

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

      7. pow1/2 73.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{{x}^{0.5}}} \]

      8. metadata-eval 73.3%

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

      9. pow-div 73.0%

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

      10. pow1 73.0%

          \[\leadsto \frac{y \cdot x}{\frac{\color{blue}{x}}{{x}^{0.5}}} \]

      11. pow1/2 73.0%

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

      12. associate-*r/ 87.0%

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

      13. pow1 87.0%

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

      14. pow1/2 87.0%

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

      15. pow-div 86.8%

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

      16. metadata-eval 86.8%

          \[\leadsto y \cdot \frac{x}{{x}^{\color{blue}{0.5}}} \]

      17. pow1/2 86.8%

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

      18. clear-num 86.9%

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

      19. pow1/2 86.9%

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

      20. pow1 86.9%

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

      21. pow-div 87.0%

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

      22. metadata-eval 87.0%

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

      23. metadata-eval 87.0%

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

      24. sqrt-pow1 87.0%

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

      25. inv-pow 87.0%

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

   6. Applied egg-rr 87.1%

      \[\leadsto \color{blue}{\frac{y}{{x}^{-0.5}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.28 \cdot 10^{+64}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+69}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{{x}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.28e+64)
   (* y (sqrt x))
   (if (<= y 2.6e+69) (- 1.0 x) (/ y (pow x -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.28e+64) {
		tmp = y * sqrt(x);
	} else if (y <= 2.6e+69) {
		tmp = 1.0 - x;
	} else {
		tmp = y / pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.28d+64)) then
        tmp = y * sqrt(x)
    else if (y <= 2.6d+69) then
        tmp = 1.0d0 - x
    else
        tmp = y / (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.28e+64) {
		tmp = y * Math.sqrt(x);
	} else if (y <= 2.6e+69) {
		tmp = 1.0 - x;
	} else {
		tmp = y / Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.28e+64:
		tmp = y * math.sqrt(x)
	elif y <= 2.6e+69:
		tmp = 1.0 - x
	else:
		tmp = y / math.pow(x, -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.28e+64)
		tmp = Float64(y * sqrt(x));
	elseif (y <= 2.6e+69)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(y / (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.28e+64)
		tmp = y * sqrt(x);
	elseif (y <= 2.6e+69)
		tmp = 1.0 - x;
	else
		tmp = y / (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.28e+64], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+69], N[(1.0 - x), $MachinePrecision], N[(y / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.28000000000000004e64

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. neg-mul-1 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around 0 92.6%

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

   if -1.28000000000000004e64 < y < 2.6000000000000002e69

   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 95.6%

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

   if 2.6000000000000002e69 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 76.7%

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

   4. Taylor expanded in x around 0 70.0%

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

      1. associate-*r* 86.9%

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

      2. sqrt-div 86.9%

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

      3. metadata-eval 86.9%

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

      4. div-inv 86.8%

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

      5. *-commutative 86.8%

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

      6. associate-*r/ 73.3%

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

      7. pow1/2 73.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{{x}^{0.5}}} \]

      8. metadata-eval 73.3%

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

      9. pow-div 73.0%

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

      10. pow1 73.0%

          \[\leadsto \frac{y \cdot x}{\frac{\color{blue}{x}}{{x}^{0.5}}} \]

      11. pow1/2 73.0%

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

      12. associate-*r/ 87.0%

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

      13. pow1 87.0%

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

      14. pow1/2 87.0%

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

      15. pow-div 86.8%

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

      16. metadata-eval 86.8%

          \[\leadsto y \cdot \frac{x}{{x}^{\color{blue}{0.5}}} \]

      17. pow1/2 86.8%

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

      18. clear-num 86.9%

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

      19. pow1/2 86.9%

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

      20. pow1 86.9%

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

      21. pow-div 87.0%

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

      22. metadata-eval 87.0%

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

      23. metadata-eval 87.0%

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

      24. sqrt-pow1 87.0%

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

      25. inv-pow 87.0%

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

   6. Applied egg-rr 87.1%

      \[\leadsto \color{blue}{\frac{y}{{x}^{-0.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.28 \cdot 10^{+64}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+69}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{{x}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+70} \lor \neg \left(y \leq 9.2 \cdot 10^{+74}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.1e+70) (not (<= y 9.2e+74))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e+70) || !(y <= 9.2e+74)) {
		tmp = y * sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.1d+70)) .or. (.not. (y <= 9.2d+74))) then
        tmp = y * sqrt(x)
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e+70) || !(y <= 9.2e+74)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.1e+70) or not (y <= 9.2e+74):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.1e+70) || !(y <= 9.2e+74))
		tmp = Float64(y * sqrt(x));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.1e+70) || ~((y <= 9.2e+74)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.1e+70], N[Not[LessEqual[y, 9.2e+74]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.10000000000000008e70 or 9.1999999999999994e74 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 96.0%

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

      1. neg-mul-1 96.0%

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

   5. Simplified 96.0%

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

   6. Taylor expanded in x around 0 90.2%

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

   if -2.10000000000000008e70 < y < 9.1999999999999994e74

   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 95.6%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+70} \lor \neg \left(y \leq 9.2 \cdot 10^{+74}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;1 + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x)))) (if (<= x 4.2e-15) (+ 1.0 t_0) (- t_0 x))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (x <= 4.2e-15) {
		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 <= 4.2d-15) 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 <= 4.2e-15) {
		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 <= 4.2e-15:
		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 <= 4.2e-15)
		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 <= 4.2e-15)
		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, 4.2e-15], N[(1.0 + t$95$0), $MachinePrecision], N[(t$95$0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.19999999999999962e-15

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.8%

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

   if 4.19999999999999962e-15 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. mul-1-neg 98.7%

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

      3. sub-neg 98.7%

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

   8. Simplified 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.35e+154) (/ (- 1.0 (* x x)) (+ 1.0 x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.35e+154) {
		tmp = (1.0 - (x * x)) / (1.0 + 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+154)) then
        tmp = (1.0d0 - (x * x)) / (1.0d0 + 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+154) {
		tmp = (1.0 - (x * x)) / (1.0 + x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.35e+154:
		tmp = (1.0 - (x * x)) / (1.0 + x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.35e+154)
		tmp = Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + 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+154)
		tmp = (1.0 - (x * x)) / (1.0 + x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.35e+154], N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35000000000000003e154

   1. Initial program 99.4%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

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

      4. sqrt-pow1 99.4%

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

      5. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y around 0 3.7%

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

      1. sub-neg 3.7%

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

      2. flip-+ 22.3%

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

      3. metadata-eval 22.3%

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

   7. Applied egg-rr 22.3%

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

   if -1.35000000000000003e154 < 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 73.0%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.2% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 4.2e-15) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.2e-15) {
		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 <= 4.2d-15) 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 <= 4.2e-15) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4.2e-15:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.2e-15)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4.2e-15)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.2e-15], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 4.19999999999999962e-15

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.8%

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

   4. Taylor expanded in y around 0 63.1%

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

   if 4.19999999999999962e-15 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.6%

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

   4. Taylor expanded in y around 0 57.3%

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

      1. neg-mul-1 57.3%

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

   6. Simplified 57.3%

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

Alternative 9: 62.7% 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 60.6%

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

Alternative 10: 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 99.8%

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

3. Taylor expanded in x around 0 69.0%

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

4. Taylor expanded in y around 0 29.6%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))