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

Percentage Accurate: 99.9% → 99.9%

Time: 5.4s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

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: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+39} \lor \neg \left(y \leq 4.8 \cdot 10^{+31}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.6e+39) (not (<= y 4.8e+31)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.6e+39) || !(y <= 4.8e+31)) {
		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 <= (-1.6d+39)) .or. (.not. (y <= 4.8d+31))) 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 <= -1.6e+39) || !(y <= 4.8e+31)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.6e+39) or not (y <= 4.8e+31):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.6e+39) || !(y <= 4.8e+31))
		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 <= -1.6e+39) || ~((y <= 4.8e+31)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.6e+39], N[Not[LessEqual[y, 4.8e+31]], $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 < -1.59999999999999996e39 or 4.79999999999999965e31 < y

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      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* 99.4%

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

      5. fma-def 99.4%

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

      6. pow1/2 99.4%

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

      7. sqrt-pow1 99.5%

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

      8. metadata-eval 99.5%

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

      9. pow1/2 99.5%

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

      10. sqrt-pow1 99.5%

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

      11. metadata-eval 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in x around 0 92.2%

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

   if -1.59999999999999996e39 < y < 4.79999999999999965e31

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+39} \lor \neg \left(y \leq 4.8 \cdot 10^{+31}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 3: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+82} \lor \neg \left(y \leq 5.2 \cdot 10^{+31}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.6e+82) (not (<= y 5.2e+31))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.6e+82) || !(y <= 5.2e+31)) {
		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 <= (-1.6d+82)) .or. (.not. (y <= 5.2d+31))) 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 <= -1.6e+82) || !(y <= 5.2e+31)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.6e+82) or not (y <= 5.2e+31):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.6e+82) || !(y <= 5.2e+31))
		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 <= -1.6e+82) || ~((y <= 5.2e+31)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.6e+82], N[Not[LessEqual[y, 5.2e+31]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.59999999999999987e82 or 5.2e31 < y

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. add-sqr-sqrt 99.2%

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

      4. associate-*l* 99.3%

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

      5. fma-def 99.4%

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

      6. pow1/2 99.4%

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

      7. sqrt-pow1 99.5%

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

      8. metadata-eval 99.5%

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

      9. pow1/2 99.5%

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

      10. sqrt-pow1 99.5%

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

      11. metadata-eval 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in y around inf 87.5%

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

   if -1.59999999999999987e82 < y < 5.2e31

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.8%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+82} \lor \neg \left(y \leq 5.2 \cdot 10^{+31}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 4: 69.8% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+136}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+131}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1e+136)
   (* x (* y (- y)))
   (if (<= y 1.7e+131) (- 1.0 x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+136) {
		tmp = x * (y * -y);
	} else if (y <= 1.7e+131) {
		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 <= (-1.1d+136)) then
        tmp = x * (y * -y)
    else if (y <= 1.7d+131) 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 <= -1.1e+136) {
		tmp = x * (y * -y);
	} else if (y <= 1.7e+131) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.1e+136:
		tmp = x * (y * -y)
	elif y <= 1.7e+131:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1e+136)
		tmp = Float64(x * Float64(y * Float64(-y)));
	elseif (y <= 1.7e+131)
		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 <= -1.1e+136)
		tmp = x * (y * -y);
	elseif (y <= 1.7e+131)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1e+136], N[(x * N[(y * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+131], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1e136

   1. Initial program 99.9%

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

      1. flip-+ 22.9%

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

      2. div-sub 22.9%

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

      3. pow2 22.9%

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

      4. associate--l- 22.9%

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

      5. *-commutative 22.9%

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

      6. *-commutative 22.9%

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

      7. swap-sqr 14.6%

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

      8. add-sqr-sqrt 14.6%

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

      9. associate--l- 14.6%

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

   3. Applied egg-rr 14.6%

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

      1. div-sub 14.6%

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

      2. associate--r+ 14.6%

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

      3. *-commutative 14.6%

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

   5. Simplified 14.6%

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

   6. Taylor expanded in y around inf 14.9%

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

      1. unpow2 14.9%

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

      2. associate-*r* 14.9%

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

      3. neg-mul-1 14.9%

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

   8. Simplified 14.9%

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

   9. Taylor expanded in x around 0 35.7%

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

       1. mul-1-neg 35.7%

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

       2. unpow2 35.7%

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

       3. distribute-rgt-neg-in 35.7%

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

       4. distribute-rgt-neg-in 35.7%

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

   11. Simplified 35.7%

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

   if -1.1e136 < y < 1.69999999999999993e131

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 87.6%

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

   if 1.69999999999999993e131 < y

   1. Initial program 99.7%

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

      1. flip-+ 27.3%

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

      2. div-sub 27.3%

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

      3. pow2 27.3%

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

      4. associate--l- 27.3%

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

      5. *-commutative 27.3%

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

      6. *-commutative 27.3%

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

      7. swap-sqr 10.3%

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

      8. add-sqr-sqrt 10.2%

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

      9. associate--l- 10.2%

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

   3. Applied egg-rr 10.2%

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

      1. div-sub 10.2%

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

      2. associate--r+ 10.2%

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

      3. *-commutative 10.2%

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

   5. Simplified 10.2%

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

   6. Taylor expanded in y around inf 10.6%

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

      1. unpow2 10.6%

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

      2. associate-*r* 10.6%

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

      3. neg-mul-1 10.6%

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

   8. Simplified 10.6%

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

   9. Taylor expanded in x around inf 31.0%

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

       1. unpow2 31.0%

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

   11. Simplified 31.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+136}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+131}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 5: 69.9% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+131}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6e+134) (* y (* y (- x))) (if (<= y 1.7e+131) (- 1.0 x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6e+134) {
		tmp = y * (y * -x);
	} else if (y <= 1.7e+131) {
		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 <= (-6d+134)) then
        tmp = y * (y * -x)
    else if (y <= 1.7d+131) 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 <= -6e+134) {
		tmp = y * (y * -x);
	} else if (y <= 1.7e+131) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6e+134:
		tmp = y * (y * -x)
	elif y <= 1.7e+131:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6e+134)
		tmp = Float64(y * Float64(y * Float64(-x)));
	elseif (y <= 1.7e+131)
		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 <= -6e+134)
		tmp = y * (y * -x);
	elseif (y <= 1.7e+131)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6e+134], N[(y * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+131], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.99999999999999993e134

   1. Initial program 99.9%

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

      1. flip-+ 22.9%

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

      2. div-sub 22.9%

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

      3. pow2 22.9%

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

      4. associate--l- 22.9%

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

      5. *-commutative 22.9%

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

      6. *-commutative 22.9%

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

      7. swap-sqr 14.6%

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

      8. add-sqr-sqrt 14.6%

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

      9. associate--l- 14.6%

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

   3. Applied egg-rr 14.6%

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

      1. div-sub 14.6%

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

      2. associate--r+ 14.6%

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

      3. *-commutative 14.6%

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

   5. Simplified 14.6%

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

   6. Taylor expanded in y around inf 14.9%

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

      1. unpow2 14.9%

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

      2. associate-*r* 14.9%

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

      3. neg-mul-1 14.9%

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

   8. Simplified 14.9%

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

   9. Taylor expanded in x around 0 35.7%

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

       1. mul-1-neg 35.7%

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

       2. unpow2 35.7%

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

       3. distribute-rgt-neg-in 35.7%

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

       4. distribute-rgt-neg-in 35.7%

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

   11. Simplified 35.7%

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

   12. Taylor expanded in x around 0 35.7%

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

       1. mul-1-neg 35.7%

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

       2. unpow2 35.7%

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

       3. associate-*r* 35.8%

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

       4. distribute-rgt-neg-in 35.8%

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

   14. Simplified 35.8%

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

   if -5.99999999999999993e134 < y < 1.69999999999999993e131

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 87.6%

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

   if 1.69999999999999993e131 < y

   1. Initial program 99.7%

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

      1. flip-+ 27.3%

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

      2. div-sub 27.3%

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

      3. pow2 27.3%

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

      4. associate--l- 27.3%

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

      5. *-commutative 27.3%

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

      6. *-commutative 27.3%

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

      7. swap-sqr 10.3%

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

      8. add-sqr-sqrt 10.2%

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

      9. associate--l- 10.2%

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

   3. Applied egg-rr 10.2%

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

      1. div-sub 10.2%

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

      2. associate--r+ 10.2%

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

      3. *-commutative 10.2%

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

   5. Simplified 10.2%

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

   6. Taylor expanded in y around inf 10.6%

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

      1. unpow2 10.6%

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

      2. associate-*r* 10.6%

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

      3. neg-mul-1 10.6%

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

   8. Simplified 10.6%

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

   9. Taylor expanded in x around inf 31.0%

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

       1. unpow2 31.0%

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

   11. Simplified 31.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+131}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 6: 66.4% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+131}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.7e+131) (- 1.0 x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.7e+131) {
		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 <= 1.7d+131) 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 <= 1.7e+131) {
		tmp = 1.0 - x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.7e+131:
		tmp = 1.0 - x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.7e+131)
		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 <= 1.7e+131)
		tmp = 1.0 - x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.7e+131], N[(1.0 - x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.69999999999999993e131

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 73.7%

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

   if 1.69999999999999993e131 < y

   1. Initial program 99.7%

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

      1. flip-+ 27.3%

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

      2. div-sub 27.3%

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

      3. pow2 27.3%

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

      4. associate--l- 27.3%

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

      5. *-commutative 27.3%

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

      6. *-commutative 27.3%

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

      7. swap-sqr 10.3%

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

      8. add-sqr-sqrt 10.2%

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

      9. associate--l- 10.2%

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

   3. Applied egg-rr 10.2%

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

      1. div-sub 10.2%

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

      2. associate--r+ 10.2%

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

      3. *-commutative 10.2%

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

   5. Simplified 10.2%

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

   6. Taylor expanded in y around inf 10.6%

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

      1. unpow2 10.6%

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

      2. associate-*r* 10.6%

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

      3. neg-mul-1 10.6%

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

   8. Simplified 10.6%

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

   9. Taylor expanded in x around inf 31.0%

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

       1. unpow2 31.0%

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

   11. Simplified 31.0%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+131}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 7: 62.3% 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 99.9%

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

   2. Taylor expanded in x around 0 67.2%

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

   if 1 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 54.0%

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

      1. mul-1-neg 54.0%

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

   4. Simplified 54.0%

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

4. Final simplification 60.3%

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

Alternative 8: 32.1% 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 33.2%

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

3. Final simplification 33.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))