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

Percentage Accurate: 99.9% → 99.9%

Time: 5.9s

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 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+63} \lor \neg \left(y \leq 9 \cdot 10^{+58}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.85e+63) (not (<= y 9e+58)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.85e+63) || !(y <= 9e+58)) {
		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 <= (-1.85d+63)) .or. (.not. (y <= 9d+58))) 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 <= -1.85e+63) || !(y <= 9e+58)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.85e+63) or not (y <= 9e+58):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.85e+63) || !(y <= 9e+58))
		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 <= -1.85e+63) || ~((y <= 9e+58)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.85e+63], N[Not[LessEqual[y, 9e+58]], $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 < -1.84999999999999984e63 or 8.9999999999999996e58 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. add-sqr-sqrt 99.3%

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

      4. associate-*l* 99.2%

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

      5. fma-def 99.2%

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

      6. pow1/2 99.2%

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

      7. sqrt-pow1 99.3%

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

      8. metadata-eval 99.3%

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

      9. pow1/2 99.3%

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

      10. sqrt-pow1 99.3%

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

      11. metadata-eval 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in x around 0 95.5%

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

   if -1.84999999999999984e63 < y < 8.9999999999999996e58

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.2%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+63} \lor \neg \left(y \leq 9 \cdot 10^{+58}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 3: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+82} \lor \neg \left(y \leq 2.3 \cdot 10^{+70}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.3e+82) (not (<= y 2.3e+70))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.3e+82) || !(y <= 2.3e+70)) {
		tmp = y * sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.3d+82)) .or. (.not. (y <= 2.3d+70))) then
        tmp = y * sqrt(x)
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.3e+82) || !(y <= 2.3e+70)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.3e+82) or not (y <= 2.3e+70):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.3e+82) || !(y <= 2.3e+70))
		tmp = Float64(y * sqrt(x));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.3e+82) || ~((y <= 2.3e+70)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.3e+82], N[Not[LessEqual[y, 2.3e+70]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2999999999999999e82 or 2.29999999999999994e70 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 91.2%

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

   if -1.2999999999999999e82 < y < 2.29999999999999994e70

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+82} \lor \neg \left(y \leq 2.3 \cdot 10^{+70}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 4: 68.3% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \left(-y \cdot y\right)\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+107}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{y}{1 + x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.7e+133)
   (* x (- (* y y)))
   (if (<= y 5.7e+107) (- 1.0 x) (* x (* y (/ y (+ 1.0 x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.7e+133) {
		tmp = x * -(y * y);
	} else if (y <= 5.7e+107) {
		tmp = 1.0 - x;
	} else {
		tmp = x * (y * (y / (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 <= (-3.7d+133)) then
        tmp = x * -(y * y)
    else if (y <= 5.7d+107) then
        tmp = 1.0d0 - x
    else
        tmp = x * (y * (y / (1.0d0 + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.7e+133) {
		tmp = x * -(y * y);
	} else if (y <= 5.7e+107) {
		tmp = 1.0 - x;
	} else {
		tmp = x * (y * (y / (1.0 + x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.7e+133:
		tmp = x * -(y * y)
	elif y <= 5.7e+107:
		tmp = 1.0 - x
	else:
		tmp = x * (y * (y / (1.0 + x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.7e+133)
		tmp = Float64(x * Float64(-Float64(y * y)));
	elseif (y <= 5.7e+107)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(x * Float64(y * Float64(y / Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.7e+133)
		tmp = x * -(y * y);
	elseif (y <= 5.7e+107)
		tmp = 1.0 - x;
	else
		tmp = x * (y * (y / (1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.7e+133], N[(x * (-N[(y * y), $MachinePrecision])), $MachinePrecision], If[LessEqual[y, 5.7e+107], N[(1.0 - x), $MachinePrecision], N[(x * N[(y * N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.70000000000000023e133

   1. Initial program 99.4%

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

      1. flip-+ 25.0%

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

      2. div-inv 25.0%

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

      3. pow2 25.0%

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

      4. *-commutative 25.0%

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

      5. *-commutative 25.0%

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

      6. swap-sqr 12.1%

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

      7. add-sqr-sqrt 12.2%

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

      8. associate--l- 12.2%

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

   3. Applied egg-rr 12.2%

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

   4. Taylor expanded in y around inf 13.0%

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

      1. mul-1-neg 13.0%

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

      2. unpow2 13.0%

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

      3. distribute-rgt-neg-out 13.0%

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

   6. Simplified 13.0%

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

   7. Taylor expanded in x around 0 21.7%

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

      1. mul-1-neg 21.7%

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

      2. unpow2 21.7%

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

      3. *-commutative 21.7%

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

      4. distribute-rgt-neg-in 21.7%

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

   9. Simplified 21.7%

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

   if -3.70000000000000023e133 < y < 5.69999999999999972e107

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 89.4%

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

   if 5.69999999999999972e107 < y

   1. Initial program 99.8%

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

      1. flip-+ 46.7%

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

      2. div-inv 46.6%

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

      3. pow2 46.6%

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

      4. *-commutative 46.6%

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

      5. *-commutative 46.6%

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

      6. swap-sqr 18.0%

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

      7. add-sqr-sqrt 17.9%

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

      8. associate--l- 17.9%

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

   3. Applied egg-rr 17.9%

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

   4. Taylor expanded in y around inf 16.6%

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

      1. mul-1-neg 16.6%

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

      2. unpow2 16.6%

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

      3. distribute-rgt-neg-out 16.6%

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

   6. Simplified 16.6%

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

   7. Taylor expanded in y around 0 17.4%

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

      1. associate-*r/ 17.4%

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

      2. mul-1-neg 17.4%

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

      3. unpow2 17.4%

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

      4. *-commutative 17.4%

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

      5. distribute-rgt-neg-in 17.4%

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

      6. associate-/l* 17.6%

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

   9. Simplified 17.6%

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

       1. expm1-log1p-u 17.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\frac{1 - x}{-y \cdot y}}\right)\right)} \]

       2. expm1-udef 17.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\frac{1 - x}{-y \cdot y}}\right)} - 1} \]

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{x}{x + 1} \cdot y\right) \cdot y\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 21.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{x}{x + 1} \cdot y\right) \cdot y\right)\right)} \]

       2. expm1-log1p 21.2%

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

       3. associate-*l/ 21.1%

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

       4. associate-/r/ 21.1%

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

       5. associate-*r/ 20.6%

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

       6. associate-/r/ 20.6%

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

       7. *-commutative 20.6%

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

   13. Simplified 20.6%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \left(-y \cdot y\right)\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+107}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{y}{1 + x}\right)\\ \end{array} \]

Alternative 5: 68.3% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \left(-y \cdot y\right)\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+107}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot y}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.05e+133)
   (* x (- (* y y)))
   (if (<= y 5.7e+107) (- 1.0 x) (* y (/ (* x y) (+ 1.0 x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.05e+133) {
		tmp = x * -(y * y);
	} else if (y <= 5.7e+107) {
		tmp = 1.0 - x;
	} else {
		tmp = y * ((x * y) / (1.0 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.05d+133)) then
        tmp = x * -(y * y)
    else if (y <= 5.7d+107) then
        tmp = 1.0d0 - x
    else
        tmp = y * ((x * y) / (1.0d0 + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.05e+133) {
		tmp = x * -(y * y);
	} else if (y <= 5.7e+107) {
		tmp = 1.0 - x;
	} else {
		tmp = y * ((x * y) / (1.0 + x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.05e+133:
		tmp = x * -(y * y)
	elif y <= 5.7e+107:
		tmp = 1.0 - x
	else:
		tmp = y * ((x * y) / (1.0 + x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.05e+133)
		tmp = Float64(x * Float64(-Float64(y * y)));
	elseif (y <= 5.7e+107)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(y * Float64(Float64(x * y) / Float64(1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.05e+133)
		tmp = x * -(y * y);
	elseif (y <= 5.7e+107)
		tmp = 1.0 - x;
	else
		tmp = y * ((x * y) / (1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.05e+133], N[(x * (-N[(y * y), $MachinePrecision])), $MachinePrecision], If[LessEqual[y, 5.7e+107], N[(1.0 - x), $MachinePrecision], N[(y * N[(N[(x * y), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.05000000000000002e133

   1. Initial program 99.4%

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

      1. flip-+ 25.0%

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

      2. div-inv 25.0%

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

      3. pow2 25.0%

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

      4. *-commutative 25.0%

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

      5. *-commutative 25.0%

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

      6. swap-sqr 12.1%

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

      7. add-sqr-sqrt 12.2%

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

      8. associate--l- 12.2%

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

   3. Applied egg-rr 12.2%

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

   4. Taylor expanded in y around inf 13.0%

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

      1. mul-1-neg 13.0%

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

      2. unpow2 13.0%

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

      3. distribute-rgt-neg-out 13.0%

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

   6. Simplified 13.0%

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

   7. Taylor expanded in x around 0 21.7%

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

      1. mul-1-neg 21.7%

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

      2. unpow2 21.7%

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

      3. *-commutative 21.7%

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

      4. distribute-rgt-neg-in 21.7%

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

   9. Simplified 21.7%

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

   if -2.05000000000000002e133 < y < 5.69999999999999972e107

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 89.4%

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

   if 5.69999999999999972e107 < y

   1. Initial program 99.8%

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

      1. flip-+ 46.7%

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

      2. div-inv 46.6%

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

      3. pow2 46.6%

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

      4. *-commutative 46.6%

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

      5. *-commutative 46.6%

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

      6. swap-sqr 18.0%

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

      7. add-sqr-sqrt 17.9%

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

      8. associate--l- 17.9%

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

   3. Applied egg-rr 17.9%

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

   4. Taylor expanded in y around inf 16.6%

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

      1. mul-1-neg 16.6%

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

      2. unpow2 16.6%

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

      3. distribute-rgt-neg-out 16.6%

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

   6. Simplified 16.6%

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

   7. Taylor expanded in y around 0 17.4%

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

      1. associate-*r/ 17.4%

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

      2. mul-1-neg 17.4%

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

      3. unpow2 17.4%

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

      4. *-commutative 17.4%

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

      5. distribute-rgt-neg-in 17.4%

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

      6. associate-/l* 17.6%

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

   9. Simplified 17.6%

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

       1. expm1-log1p-u 17.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\frac{1 - x}{-y \cdot y}}\right)\right)} \]

       2. expm1-udef 17.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\frac{1 - x}{-y \cdot y}}\right)} - 1} \]

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{x}{x + 1} \cdot y\right) \cdot y\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 21.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{x}{x + 1} \cdot y\right) \cdot y\right)\right)} \]

       2. expm1-log1p 21.2%

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

       3. *-commutative 21.2%

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

       4. associate-*l/ 21.1%

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

   13. Simplified 21.1%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \left(-y \cdot y\right)\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+107}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot y}{1 + x}\\ \end{array} \]

Alternative 6: 68.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+132}:\\ \;\;\;\;y \cdot \left(-y \cdot \frac{x}{1 + x}\right)\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+107}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot y}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1e+132)
   (* y (- (* y (/ x (+ 1.0 x)))))
   (if (<= y 5.7e+107) (- 1.0 x) (* y (/ (* x y) (+ 1.0 x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+132) {
		tmp = y * -(y * (x / (1.0 + x)));
	} else if (y <= 5.7e+107) {
		tmp = 1.0 - x;
	} else {
		tmp = y * ((x * y) / (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 <= (-1.1d+132)) then
        tmp = y * -(y * (x / (1.0d0 + x)))
    else if (y <= 5.7d+107) then
        tmp = 1.0d0 - x
    else
        tmp = y * ((x * y) / (1.0d0 + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+132) {
		tmp = y * -(y * (x / (1.0 + x)));
	} else if (y <= 5.7e+107) {
		tmp = 1.0 - x;
	} else {
		tmp = y * ((x * y) / (1.0 + x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.1e+132:
		tmp = y * -(y * (x / (1.0 + x)))
	elif y <= 5.7e+107:
		tmp = 1.0 - x
	else:
		tmp = y * ((x * y) / (1.0 + x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1e+132)
		tmp = Float64(y * Float64(-Float64(y * Float64(x / Float64(1.0 + x)))));
	elseif (y <= 5.7e+107)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(y * Float64(Float64(x * y) / Float64(1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.1e+132)
		tmp = y * -(y * (x / (1.0 + x)));
	elseif (y <= 5.7e+107)
		tmp = 1.0 - x;
	else
		tmp = y * ((x * y) / (1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1e+132], N[(y * (-N[(y * N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[y, 5.7e+107], N[(1.0 - x), $MachinePrecision], N[(y * N[(N[(x * y), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.09999999999999994e132

   1. Initial program 99.4%

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

      1. flip-+ 25.0%

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

      2. div-inv 25.0%

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

      3. pow2 25.0%

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

      4. *-commutative 25.0%

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

      5. *-commutative 25.0%

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

      6. swap-sqr 12.1%

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

      7. add-sqr-sqrt 12.2%

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

      8. associate--l- 12.2%

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

   3. Applied egg-rr 12.2%

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

   4. Taylor expanded in y around inf 13.0%

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

      1. mul-1-neg 13.0%

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

      2. unpow2 13.0%

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

      3. distribute-rgt-neg-out 13.0%

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

   6. Simplified 13.0%

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

   7. Taylor expanded in y around 0 2.7%

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

      1. associate-*r/ 2.7%

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

      2. mul-1-neg 2.7%

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

      3. unpow2 2.7%

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

      4. *-commutative 2.7%

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

      5. distribute-rgt-neg-in 2.7%

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

      6. associate-/l* 2.7%

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

   9. Simplified 2.7%

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

       1. associate-/r/ 2.7%

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

       2. distribute-lft-neg-in 2.7%

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

       3. associate-*r* 3.0%

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

       4. *-un-lft-identity 3.0%

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

       5. *-un-lft-identity 3.0%

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

       6. sub-neg 3.0%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 8.0%

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

       9. sqr-neg 8.0%

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

       10. sqrt-unprod 22.0%

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

       11. add-sqr-sqrt 22.0%

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

       12. +-commutative 22.0%

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

   11. Applied egg-rr 22.0%

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

   if -1.09999999999999994e132 < y < 5.69999999999999972e107

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 89.4%

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

   if 5.69999999999999972e107 < y

   1. Initial program 99.8%

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

      1. flip-+ 46.7%

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

      2. div-inv 46.6%

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

      3. pow2 46.6%

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

      4. *-commutative 46.6%

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

      5. *-commutative 46.6%

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

      6. swap-sqr 18.0%

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

      7. add-sqr-sqrt 17.9%

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

      8. associate--l- 17.9%

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

   3. Applied egg-rr 17.9%

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

   4. Taylor expanded in y around inf 16.6%

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

      1. mul-1-neg 16.6%

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

      2. unpow2 16.6%

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

      3. distribute-rgt-neg-out 16.6%

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

   6. Simplified 16.6%

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

   7. Taylor expanded in y around 0 17.4%

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

      1. associate-*r/ 17.4%

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

      2. mul-1-neg 17.4%

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

      3. unpow2 17.4%

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

      4. *-commutative 17.4%

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

      5. distribute-rgt-neg-in 17.4%

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

      6. associate-/l* 17.6%

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

   9. Simplified 17.6%

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

       1. expm1-log1p-u 17.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\frac{1 - x}{-y \cdot y}}\right)\right)} \]

       2. expm1-udef 17.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\frac{1 - x}{-y \cdot y}}\right)} - 1} \]

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{x}{x + 1} \cdot y\right) \cdot y\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 21.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{x}{x + 1} \cdot y\right) \cdot y\right)\right)} \]

       2. expm1-log1p 21.2%

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

       3. *-commutative 21.2%

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

       4. associate-*l/ 21.1%

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

   13. Simplified 21.1%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+132}:\\ \;\;\;\;y \cdot \left(-y \cdot \frac{x}{1 + x}\right)\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+107}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot y}{1 + x}\\ \end{array} \]

Alternative 7: 68.5% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+130}:\\ \;\;\;\;x \cdot \left(-y \cdot y\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+166}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.6e+130) (* x (- (* y y))) (if (<= y 2e+166) (- 1.0 x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.6e+130) {
		tmp = x * -(y * y);
	} else if (y <= 2e+166) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8.6d+130)) then
        tmp = x * -(y * y)
    else if (y <= 2d+166) then
        tmp = 1.0d0 - x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.6e+130) {
		tmp = x * -(y * y);
	} else if (y <= 2e+166) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.6e+130:
		tmp = x * -(y * y)
	elif y <= 2e+166:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.6e+130)
		tmp = Float64(x * Float64(-Float64(y * y)));
	elseif (y <= 2e+166)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.6e+130)
		tmp = x * -(y * y);
	elseif (y <= 2e+166)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.6e+130], N[(x * (-N[(y * y), $MachinePrecision])), $MachinePrecision], If[LessEqual[y, 2e+166], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.59999999999999968e130

   1. Initial program 99.4%

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

      1. flip-+ 25.0%

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

      2. div-inv 25.0%

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

      3. pow2 25.0%

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

      4. *-commutative 25.0%

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

      5. *-commutative 25.0%

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

      6. swap-sqr 12.1%

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

      7. add-sqr-sqrt 12.2%

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

      8. associate--l- 12.2%

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

   3. Applied egg-rr 12.2%

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

   4. Taylor expanded in y around inf 13.0%

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

      1. mul-1-neg 13.0%

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

      2. unpow2 13.0%

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

      3. distribute-rgt-neg-out 13.0%

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

   6. Simplified 13.0%

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

   7. Taylor expanded in x around 0 21.7%

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

      1. mul-1-neg 21.7%

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

      2. unpow2 21.7%

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

      3. *-commutative 21.7%

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

      4. distribute-rgt-neg-in 21.7%

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

   9. Simplified 21.7%

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

   if -8.59999999999999968e130 < y < 1.99999999999999988e166

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 83.3%

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

   if 1.99999999999999988e166 < y

   1. Initial program 99.8%

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

      1. flip-+ 33.0%

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

      2. div-inv 33.0%

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

      3. pow2 33.0%

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

      4. *-commutative 33.0%

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

      5. *-commutative 33.0%

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

      6. swap-sqr 3.7%

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

      7. add-sqr-sqrt 3.7%

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

      8. associate--l- 3.7%

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

   3. Applied egg-rr 3.7%

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

   4. Taylor expanded in y around inf 4.9%

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

      1. mul-1-neg 4.9%

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

      2. unpow2 4.9%

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

      3. distribute-rgt-neg-out 4.9%

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

   6. Simplified 4.9%

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

   7. Taylor expanded in x around inf 28.0%

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

      1. unpow2 28.0%

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

   9. Simplified 28.0%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+130}:\\ \;\;\;\;x \cdot \left(-y \cdot y\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+166}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 8: 65.3% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{+166}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.75e+166) (- 1.0 x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.75e+166) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.75d+166) then
        tmp = 1.0d0 - x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.75e+166) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.75e+166:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.75e+166)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.75e+166)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.75e+166], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.7499999999999999e166

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 70.4%

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

   if 1.7499999999999999e166 < y

   1. Initial program 99.8%

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

      1. flip-+ 33.0%

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

      2. div-inv 33.0%

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

      3. pow2 33.0%

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

      4. *-commutative 33.0%

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

      5. *-commutative 33.0%

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

      6. swap-sqr 3.7%

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

      7. add-sqr-sqrt 3.7%

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

      8. associate--l- 3.7%

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

   3. Applied egg-rr 3.7%

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

   4. Taylor expanded in y around inf 4.9%

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

      1. mul-1-neg 4.9%

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

      2. unpow2 4.9%

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

      3. distribute-rgt-neg-out 4.9%

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

   6. Simplified 4.9%

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

   7. Taylor expanded in x around inf 28.0%

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

      1. unpow2 28.0%

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

   9. Simplified 28.0%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{+166}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 9: 60.8% accurate, 26.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 62.0%

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

   if 1 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 60.6%

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

      1. mul-1-neg 60.6%

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

   4. Simplified 60.6%

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

4. Final simplification 61.3%

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

Alternative 10: 30.7% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 33.7%

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

3. Final simplification 33.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023192 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))