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

Percentage Accurate: 99.9% → 99.9%

Time: 9.0s

Alternatives: 11

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

\[\begin{array}{l} \\ \frac{y}{{x}^{-0.5}} + \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (/ y (pow x -0.5)) (- 1.0 x)))

C

double code(double x, double y) {
	return (y / pow(x, -0.5)) + (1.0 - x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y / (x ** (-0.5d0))) + (1.0d0 - x)
end function

Java

public static double code(double x, double y) {
	return (y / Math.pow(x, -0.5)) + (1.0 - x);
}

Python

def code(x, y):
	return (y / math.pow(x, -0.5)) + (1.0 - x)

Julia

function code(x, y)
	return Float64(Float64(y / (x ^ -0.5)) + Float64(1.0 - x))
end

MATLAB

function tmp = code(x, y)
	tmp = (y / (x ^ -0.5)) + (1.0 - x);
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf 90.1%

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

   1. associate--l+ 90.1%

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

   2. distribute-rgt-in 90.1%

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

5. Simplified 99.8%

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

   1. sqrt-div 99.8%

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

   2. metadata-eval 99.8%

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

   3. un-div-inv 99.8%

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

7. Applied egg-rr 99.8%

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

   1. frac-2neg 99.8%

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

   2. clear-num 99.8%

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

   3. frac-2neg 99.8%

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

   4. pow1/2 99.8%

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

   5. pow1 99.8%

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

   6. pow-div 99.8%

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

   7. metadata-eval 99.8%

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

   8. metadata-eval 99.8%

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

   9. sqrt-pow1 99.8%

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

   10. inv-pow 99.8%

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

   11. associate-*l/ 99.9%

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

   12. *-un-lft-identity 99.9%

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

   13. inv-pow 99.9%

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

   14. sqrt-pow1 99.9%

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

   15. metadata-eval 99.9%

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

9. Applied egg-rr 99.9%

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

Alternative 2: 94.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+36} \lor \neg \left(y \leq 2.5 \cdot 10^{+37}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.2e+36) (not (<= y 2.5e+37)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.2e+36) || !(y <= 2.5e+37)) {
		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 <= (-3.2d+36)) .or. (.not. (y <= 2.5d+37))) 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 <= -3.2e+36) || !(y <= 2.5e+37)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.2e+36) or not (y <= 2.5e+37):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.2e+36) || !(y <= 2.5e+37))
		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 <= -3.2e+36) || ~((y <= 2.5e+37)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.2e+36], N[Not[LessEqual[y, 2.5e+37]], $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 < -3.1999999999999999e36 or 2.49999999999999994e37 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 95.9%

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

   if -3.1999999999999999e36 < y < 2.49999999999999994e37

   1. Initial program 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \left(1 - x\right) + \color{blue}{\left(\sqrt[3]{y \cdot \sqrt{x}} \cdot \sqrt[3]{y \cdot \sqrt{x}}\right) \cdot \sqrt[3]{y \cdot \sqrt{x}}} \]

      2. pow3 100.0%

         \[\leadsto \left(1 - x\right) + \color{blue}{{\left(\sqrt[3]{y \cdot \sqrt{x}}\right)}^{3}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left(1 - x\right) + \color{blue}{{\left(\sqrt[3]{y \cdot \sqrt{x}}\right)}^{3}} \]

   5. Taylor expanded in y around 0 99.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+36} \lor \neg \left(y \leq 2.5 \cdot 10^{+37}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.06 \cdot 10^{+49} \lor \neg \left(y \leq 2.2 \cdot 10^{+73}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.06e+49) (not (<= y 2.2e+73))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.06e+49) || !(y <= 2.2e+73)) {
		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 <= (-2.06d+49)) .or. (.not. (y <= 2.2d+73))) 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 <= -2.06e+49) || !(y <= 2.2e+73)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.06e+49) or not (y <= 2.2e+73):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.06e+49) || !(y <= 2.2e+73))
		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 <= -2.06e+49) || ~((y <= 2.2e+73)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.06e+49], N[Not[LessEqual[y, 2.2e+73]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0600000000000001e49 or 2.2e73 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 78.7%

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

      1. associate--l+ 78.7%

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

      2. distribute-rgt-in 78.7%

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

   5. Simplified 99.6%

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

      1. sqrt-div 99.5%

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

      2. metadata-eval 99.5%

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

      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 99.5%

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

      3. frac-2neg 99.5%

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

      4. pow1/2 99.5%

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

      5. pow1 99.5%

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

      6. pow-div 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

      9. sqrt-pow1 99.6%

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

      10. inv-pow 99.6%

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

      11. associate-*l/ 99.7%

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

      12. *-un-lft-identity 99.7%

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

      13. inv-pow 99.7%

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

      14. sqrt-pow1 99.7%

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

      15. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in y around inf 91.8%

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

   if -2.0600000000000001e49 < y < 2.2e73

   1. Initial program 100.0%

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

      1. add-cube-cbrt 99.9%

         \[\leadsto \left(1 - x\right) + \color{blue}{\left(\sqrt[3]{y \cdot \sqrt{x}} \cdot \sqrt[3]{y \cdot \sqrt{x}}\right) \cdot \sqrt[3]{y \cdot \sqrt{x}}} \]

      2. pow3 99.9%

         \[\leadsto \left(1 - x\right) + \color{blue}{{\left(\sqrt[3]{y \cdot \sqrt{x}}\right)}^{3}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left(1 - x\right) + \color{blue}{{\left(\sqrt[3]{y \cdot \sqrt{x}}\right)}^{3}} \]

   5. Taylor expanded in y around 0 97.3%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.06 \cdot 10^{+49} \lor \neg \left(y \leq 2.2 \cdot 10^{+73}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(y / (x ^ -0.5)) + 1.0);
	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 = (y / (x ^ -0.5)) + 1.0;
	else
		tmp = (y * sqrt(x)) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.0], N[(N[(y / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] + 1.0), $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 80.5%

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

      1. associate--l+ 80.5%

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

      2. distribute-rgt-in 80.5%

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

   5. Simplified 99.8%

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

      1. sqrt-div 99.8%

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

      2. metadata-eval 99.8%

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

      3. un-div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. frac-2neg 99.8%

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

      2. clear-num 99.8%

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

      3. frac-2neg 99.8%

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

      4. pow1/2 99.8%

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

      5. pow1 99.8%

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

      6. pow-div 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

      9. sqrt-pow1 99.8%

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

      10. inv-pow 99.8%

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

      11. associate-*l/ 99.9%

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

      12. *-un-lft-identity 99.9%

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

      13. inv-pow 99.9%

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

      14. sqrt-pow1 99.9%

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

      15. metadata-eval 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around 0 98.1%

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

   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 99.9%

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

      1. neg-mul-1 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.0%

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

Alternative 5: 98.7% 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 98.0%

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

   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 99.9%

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

      1. neg-mul-1 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.0%

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

Alternative 6: 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 7: 67.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{y \cdot \left(x + -1\right)}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x + 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.2e+38) (/ (* y (+ x -1.0)) (- y)) (/ (* y (+ x 1.0)) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.2e+38) {
		tmp = (y * (x + -1.0)) / -y;
	} else {
		tmp = (y * (x + 1.0)) / 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 <= 4.2d+38) then
        tmp = (y * (x + (-1.0d0))) / -y
    else
        tmp = (y * (x + 1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.2e+38) {
		tmp = (y * (x + -1.0)) / -y;
	} else {
		tmp = (y * (x + 1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.2e+38:
		tmp = (y * (x + -1.0)) / -y
	else:
		tmp = (y * (x + 1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.2e+38)
		tmp = Float64(Float64(y * Float64(x + -1.0)) / Float64(-y));
	else
		tmp = Float64(Float64(y * Float64(x + 1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.2e+38)
		tmp = (y * (x + -1.0)) / -y;
	else
		tmp = (y * (x + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.2e+38], N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / (-y)), $MachinePrecision], N[(N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2e38

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 84.3%

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

      1. associate--l+ 84.3%

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

      2. div-sub 84.3%

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

   5. Simplified 84.3%

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

   6. Taylor expanded in y around 0 52.3%

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

      1. *-commutative 52.3%

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

      2. frac-2neg 52.3%

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

      3. associate-*l/ 72.8%

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

      4. sub-neg 72.8%

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

      5. distribute-neg-in 72.8%

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

      6. metadata-eval 72.8%

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

      7. remove-double-neg 72.8%

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

   8. Applied egg-rr 72.8%

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

   if 4.2e38 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf 99.5%

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

      1. associate--l+ 99.5%

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

      2. div-sub 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around 0 15.6%

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

      1. associate-*r/ 11.0%

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

      2. sub-neg 11.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 26.7%

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

      5. sqr-neg 26.7%

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

      6. sqrt-unprod 26.7%

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

      7. add-sqr-sqrt 26.7%

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

      8. +-commutative 26.7%

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

   8. Applied egg-rr 26.7%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{y \cdot \left(x + -1\right)}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x + 1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+130}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x + 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8e+130) (- 1.0 x) (/ (* y (+ x 1.0)) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e+130) {
		tmp = 1.0 - x;
	} else {
		tmp = (y * (x + 1.0)) / 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 <= 8d+130) then
        tmp = 1.0d0 - x
    else
        tmp = (y * (x + 1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8e+130) {
		tmp = 1.0 - x;
	} else {
		tmp = (y * (x + 1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e+130:
		tmp = 1.0 - x
	else:
		tmp = (y * (x + 1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e+130)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(y * Float64(x + 1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8e+130)
		tmp = 1.0 - x;
	else
		tmp = (y * (x + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e+130], N[(1.0 - x), $MachinePrecision], N[(N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.0000000000000005e130

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.5%

         \[\leadsto \left(1 - x\right) + \color{blue}{\left(\sqrt[3]{y \cdot \sqrt{x}} \cdot \sqrt[3]{y \cdot \sqrt{x}}\right) \cdot \sqrt[3]{y \cdot \sqrt{x}}} \]

      2. pow3 99.5%

         \[\leadsto \left(1 - x\right) + \color{blue}{{\left(\sqrt[3]{y \cdot \sqrt{x}}\right)}^{3}} \]

   4. Applied egg-rr 99.5%

      \[\leadsto \left(1 - x\right) + \color{blue}{{\left(\sqrt[3]{y \cdot \sqrt{x}}\right)}^{3}} \]

   5. Taylor expanded in y around 0 66.1%

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

   if 8.0000000000000005e130 < 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 2.4%

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

      1. associate-*r/ 2.3%

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

      2. sub-neg 2.3%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 24.3%

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

      5. sqr-neg 24.3%

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

      6. sqrt-unprod 24.3%

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

      7. add-sqr-sqrt 24.3%

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

      8. +-commutative 24.3%

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

   8. Applied egg-rr 24.3%

      \[\leadsto \color{blue}{\frac{y \cdot \left(x + 1\right)}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 61.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 2.2e+16) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.2e+16) {
		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 <= 2.2d+16) 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 <= 2.2e+16) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.2e+16:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.2e+16)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.2e+16)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.2e+16], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 2.2e16

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 98.1%

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

   4. Taylor expanded in y around 0 54.0%

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

   if 2.2e16 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 75.0%

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

      1. associate--l+ 75.0%

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

      2. div-sub 75.0%

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

   5. Simplified 75.0%

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

   6. Taylor expanded in y around 0 31.4%

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

   7. Taylor expanded in x around inf 56.2%

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

      1. mul-1-neg 56.2%

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

   9. Simplified 56.2%

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

Alternative 10: 63.4% 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-cube-cbrt 99.3%

      \[\leadsto \left(1 - x\right) + \color{blue}{\left(\sqrt[3]{y \cdot \sqrt{x}} \cdot \sqrt[3]{y \cdot \sqrt{x}}\right) \cdot \sqrt[3]{y \cdot \sqrt{x}}} \]

   2. pow3 99.3%

      \[\leadsto \left(1 - x\right) + \color{blue}{{\left(\sqrt[3]{y \cdot \sqrt{x}}\right)}^{3}} \]

4. Applied egg-rr 99.3%

   \[\leadsto \left(1 - x\right) + \color{blue}{{\left(\sqrt[3]{y \cdot \sqrt{x}}\right)}^{3}} \]

5. Taylor expanded in y around 0 55.9%

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

Alternative 11: 31.4% 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 72.6%

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

4. Taylor expanded in y around 0 28.8%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))