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

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf 93.1%

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

   1. associate--l+ 93.1%

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

   2. distribute-rgt-in 93.1%

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

5. Simplified 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-+l- 99.8%

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

   3. *-commutative 99.8%

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

   4. sqrt-div 99.8%

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

   5. metadata-eval 99.8%

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

   6. un-div-inv 99.8%

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

7. Applied egg-rr 99.8%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.9%

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

   3. pow1/2 99.9%

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

   4. pow1 99.9%

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

   5. pow-div 99.9%

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

   6. metadata-eval 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 94.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+18} \lor \neg \left(y \leq 2.1 \cdot 10^{+35}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.2e+18) (not (<= y 2.1e+35)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.2e+18) || !(y <= 2.1e+35)) {
		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 <= (-7.2d+18)) .or. (.not. (y <= 2.1d+35))) 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 <= -7.2e+18) || !(y <= 2.1e+35)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.2e+18) or not (y <= 2.1e+35):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.2e+18) || !(y <= 2.1e+35))
		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 <= -7.2e+18) || ~((y <= 2.1e+35)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.2e+18], N[Not[LessEqual[y, 2.1e+35]], $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 < -7.2e18 or 2.0999999999999999e35 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 95.1%

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

   if -7.2e18 < y < 2.0999999999999999e35

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. pow2 100.0%

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

      3. pow1/2 100.0%

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

      4. sqrt-pow1 100.0%

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

      5. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 99.6%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+18} \lor \neg \left(y \leq 2.1 \cdot 10^{+35}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+52} \lor \neg \left(y \leq 2.4 \cdot 10^{+45}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7e+52) (not (<= y 2.4e+45))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7e+52) || !(y <= 2.4e+45)) {
		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 <= (-7d+52)) .or. (.not. (y <= 2.4d+45))) 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 <= -7e+52) || !(y <= 2.4e+45)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7e+52) or not (y <= 2.4e+45):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7e+52) || !(y <= 2.4e+45))
		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 <= -7e+52) || ~((y <= 2.4e+45)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7e+52], N[Not[LessEqual[y, 2.4e+45]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7e52 or 2.39999999999999989e45 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 84.5%

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

      1. associate--l+ 84.5%

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

      2. distribute-rgt-in 84.5%

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

   5. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-+l- 99.5%

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

      3. *-commutative 99.5%

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

      4. sqrt-div 99.5%

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

      5. metadata-eval 99.5%

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

      6. un-div-inv 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.7%

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

      3. pow1/2 99.7%

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

      4. pow1 99.7%

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

      5. pow-div 99.8%

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

      6. metadata-eval 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in y around inf 89.8%

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

   if -7e52 < y < 2.39999999999999989e45

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. pow2 100.0%

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

      3. pow1/2 100.0%

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

      4. sqrt-pow1 100.0%

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

      5. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 97.5%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+52} \lor \neg \left(y \leq 2.4 \cdot 10^{+45}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 0.096:\\ \;\;\;\;1 + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x)))) (if (<= x 0.096) (+ 1.0 t_0) (- t_0 x))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (x <= 0.096) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(x)
    if (x <= 0.096d0) then
        tmp = 1.0d0 + t_0
    else
        tmp = t_0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * Math.sqrt(x);
	double tmp;
	if (x <= 0.096) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * math.sqrt(x)
	tmp = 0
	if x <= 0.096:
		tmp = 1.0 + t_0
	else:
		tmp = t_0 - x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * sqrt(x))
	tmp = 0.0
	if (x <= 0.096)
		tmp = Float64(1.0 + t_0);
	else
		tmp = Float64(t_0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * sqrt(x);
	tmp = 0.0;
	if (x <= 0.096)
		tmp = 1.0 + t_0;
	else
		tmp = t_0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.096], N[(1.0 + t$95$0), $MachinePrecision], N[(t$95$0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.096000000000000002

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.0%

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

   if 0.096000000000000002 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.6%

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

   4. Taylor expanded in x around 0 98.7%

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

      1. neg-mul-1 98.7%

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

      2. +-commutative 98.7%

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

      3. fma-define 98.7%

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

      4. fmm-def 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.096:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 6: 66.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - x \cdot x\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{+156}:\\ \;\;\;\;\frac{t\_0}{1 + x}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+35}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1 + \frac{1}{\frac{-1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x x))))
   (if (<= y -1.15e+156)
     (/ t_0 (+ 1.0 x))
     (if (<= y 7.5e+35) (- 1.0 x) (/ t_0 (+ 1.0 (/ 1.0 (/ -1.0 x))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x * x);
	double tmp;
	if (y <= -1.15e+156) {
		tmp = t_0 / (1.0 + x);
	} else if (y <= 7.5e+35) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0 / (1.0 + (1.0 / (-1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x * x)
    if (y <= (-1.15d+156)) then
        tmp = t_0 / (1.0d0 + x)
    else if (y <= 7.5d+35) then
        tmp = 1.0d0 - x
    else
        tmp = t_0 / (1.0d0 + (1.0d0 / ((-1.0d0) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (x * x);
	double tmp;
	if (y <= -1.15e+156) {
		tmp = t_0 / (1.0 + x);
	} else if (y <= 7.5e+35) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0 / (1.0 + (1.0 / (-1.0 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x * x)
	tmp = 0
	if y <= -1.15e+156:
		tmp = t_0 / (1.0 + x)
	elif y <= 7.5e+35:
		tmp = 1.0 - x
	else:
		tmp = t_0 / (1.0 + (1.0 / (-1.0 / x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x * x))
	tmp = 0.0
	if (y <= -1.15e+156)
		tmp = Float64(t_0 / Float64(1.0 + x));
	elseif (y <= 7.5e+35)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(t_0 / Float64(1.0 + Float64(1.0 / Float64(-1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x * x);
	tmp = 0.0;
	if (y <= -1.15e+156)
		tmp = t_0 / (1.0 + x);
	elseif (y <= 7.5e+35)
		tmp = 1.0 - x;
	else
		tmp = t_0 / (1.0 + (1.0 / (-1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.15e+156], N[(t$95$0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+35], N[(1.0 - x), $MachinePrecision], N[(t$95$0 / N[(1.0 + N[(1.0 / N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1499999999999999e156

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 99.6%

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

      2. pow2 99.6%

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

      3. pow1/2 99.6%

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

      4. sqrt-pow1 99.6%

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

      5. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0 4.5%

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

      1. sub-neg 4.5%

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

      2. flip-+ 26.7%

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

      3. metadata-eval 26.7%

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

   7. Applied egg-rr 26.7%

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

   8. Taylor expanded in x around 0 26.7%

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

   if -1.1499999999999999e156 < y < 7.4999999999999999e35

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. pow2 99.9%

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

      3. pow1/2 99.9%

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

      4. sqrt-pow1 99.9%

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

      5. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 84.8%

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

   if 7.4999999999999999e35 < y

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

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

      4. sqrt-pow1 99.4%

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

      5. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y around 0 9.8%

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

      1. sub-neg 9.8%

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

      2. flip-+ 9.8%

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

      3. metadata-eval 9.8%

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

   7. Applied egg-rr 9.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 11.1%

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

      3. sqr-neg 11.1%

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

      4. sqrt-unprod 25.6%

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

      5. pow2 25.6%

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

      6. pow1/2 25.6%

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

      7. metadata-eval 25.6%

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

      8. pow-div 25.6%

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

      9. pow1 25.6%

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

      10. pow1/2 25.6%

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

      11. pow2 25.6%

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

      12. clear-num 25.6%

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

      13. clear-num 25.6%

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

      14. frac-times 25.6%

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

      15. metadata-eval 25.6%

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

      16. pow1/2 25.6%

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

      17. pow1 25.6%

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

      18. pow-div 25.6%

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

      19. metadata-eval 25.6%

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

      20. pow1/2 25.6%

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

      21. pow1 25.6%

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

      22. pow-div 25.6%

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

      23. metadata-eval 25.6%

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

   9. Applied egg-rr 25.6%

      \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \color{blue}{\frac{1}{{x}^{-0.5} \cdot {x}^{-0.5}}}} \]
   10. Step-by-step derivation

       1. pow-sqr 25.6%

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

       2. metadata-eval 25.6%

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

       3. unpow-1 25.6%

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

   11. Simplified 25.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+156}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+35}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + \frac{1}{\frac{-1}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+148}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+45}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e+148)
   (/ (- 1.0 (* x x)) (+ 1.0 x))
   (if (<= y 2.4e+45) (- 1.0 x) (/ (+ 1.0 (* x x)) (+ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05e+148) {
		tmp = (1.0 - (x * x)) / (1.0 + x);
	} else if (y <= 2.4e+45) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 + (x * x)) / (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 <= (-1.05d+148)) then
        tmp = (1.0d0 - (x * x)) / (1.0d0 + x)
    else if (y <= 2.4d+45) then
        tmp = 1.0d0 - x
    else
        tmp = (1.0d0 + (x * x)) / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.05e+148) {
		tmp = (1.0 - (x * x)) / (1.0 + x);
	} else if (y <= 2.4e+45) {
		tmp = 1.0 - x;
	} else {
		tmp = (1.0 + (x * x)) / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.05e+148:
		tmp = (1.0 - (x * x)) / (1.0 + x)
	elif y <= 2.4e+45:
		tmp = 1.0 - x
	else:
		tmp = (1.0 + (x * x)) / (1.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05e+148)
		tmp = Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + x));
	elseif (y <= 2.4e+45)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(Float64(1.0 + Float64(x * x)) / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.05e+148)
		tmp = (1.0 - (x * x)) / (1.0 + x);
	elseif (y <= 2.4e+45)
		tmp = 1.0 - x;
	else
		tmp = (1.0 + (x * x)) / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05e+148], N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e+45], N[(1.0 - x), $MachinePrecision], N[(N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.04999999999999999e148

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 99.5%

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

      2. pow2 99.5%

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

      3. pow1/2 99.5%

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

      4. sqrt-pow1 99.5%

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

      5. metadata-eval 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0 4.4%

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

      1. sub-neg 4.4%

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

      2. flip-+ 24.5%

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

      3. metadata-eval 24.5%

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

   7. Applied egg-rr 24.5%

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

   8. Taylor expanded in x around 0 24.5%

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

   if -1.04999999999999999e148 < y < 2.39999999999999989e45

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 99.9%

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

      2. pow2 99.9%

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

      3. pow1/2 99.9%

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

      4. sqrt-pow1 99.9%

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

      5. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 86.3%

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

   if 2.39999999999999989e45 < y

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

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

      4. sqrt-pow1 99.4%

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

      5. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y around 0 7.1%

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

      1. sub-neg 7.1%

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

      2. flip-+ 7.0%

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

      3. metadata-eval 7.0%

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

   7. Applied egg-rr 7.0%

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

      1. neg-sub0 7.0%

         \[\leadsto \frac{1 - \color{blue}{\left(0 - x\right)} \cdot \left(-x\right)}{1 - \left(-x\right)} \]

      2. sub-neg 7.0%

         \[\leadsto \frac{1 - \color{blue}{\left(0 + \left(-x\right)\right)} \cdot \left(-x\right)}{1 - \left(-x\right)} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \frac{1 - \left(0 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right) \cdot \left(-x\right)}{1 - \left(-x\right)} \]

      4. sqrt-unprod 23.4%

         \[\leadsto \frac{1 - \left(0 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right) \cdot \left(-x\right)}{1 - \left(-x\right)} \]

      5. sqr-neg 23.4%

         \[\leadsto \frac{1 - \left(0 + \sqrt{\color{blue}{x \cdot x}}\right) \cdot \left(-x\right)}{1 - \left(-x\right)} \]

      6. sqrt-unprod 23.4%

         \[\leadsto \frac{1 - \left(0 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) \cdot \left(-x\right)}{1 - \left(-x\right)} \]

      7. add-sqr-sqrt 23.4%

         \[\leadsto \frac{1 - \left(0 + \color{blue}{x}\right) \cdot \left(-x\right)}{1 - \left(-x\right)} \]

   9. Applied egg-rr 23.4%

      \[\leadsto \frac{1 - \color{blue}{\left(0 + x\right)} \cdot \left(-x\right)}{1 - \left(-x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+148}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+45}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.0% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.35d+154)) then
        tmp = (1.0d0 - (x * x)) / (1.0d0 + x)
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1.35000000000000003e154

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 99.6%

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

      2. pow2 99.6%

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

      3. pow1/2 99.6%

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

      4. sqrt-pow1 99.6%

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

      5. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0 4.5%

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

      1. sub-neg 4.5%

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

      2. flip-+ 26.0%

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

      3. metadata-eval 26.0%

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

   7. Applied egg-rr 26.0%

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

   8. Taylor expanded in x around 0 26.0%

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

   if -1.35000000000000003e154 < y

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 99.7%

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

      2. pow2 99.7%

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

      3. pow1/2 99.7%

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

      4. sqrt-pow1 99.8%

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

      5. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 66.2%

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

4. Final simplification 61.5%

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

Alternative 9: 61.6% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.096:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 0.096) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.096) {
		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 <= 0.096d0) 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 <= 0.096) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.096:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.096)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.096)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.096], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 0.096000000000000002

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.0%

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

   4. Taylor expanded in y around 0 64.1%

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

   if 0.096000000000000002 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.6%

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

   4. Taylor expanded in y around 0 52.5%

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

      1. neg-mul-1 52.5%

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

   6. Simplified 52.5%

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

Alternative 10: 62.8% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. add-sqr-sqrt 99.7%

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

   2. pow2 99.7%

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

   3. pow1/2 99.7%

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

   4. sqrt-pow1 99.7%

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

   5. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in y around 0 59.0%

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

Alternative 11: 32.0% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 71.9%

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

4. Taylor expanded in y around 0 31.3%

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

Reproduce

herbie shell --seed 2024186 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))