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

Percentage Accurate: 99.9% → 99.9%

Time: 8.6s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 95.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+55} \lor \neg \left(y \leq 4.5 \cdot 10^{+27}\right):\\ \;\;\;\;\sqrt{x} \cdot y + 1\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8e+55) (not (<= y 4.5e+27))) (+ (* (sqrt x) y) 1.0) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8e+55) || !(y <= 4.5e+27)) {
		tmp = (sqrt(x) * y) + 1.0;
	} 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 <= (-8d+55)) .or. (.not. (y <= 4.5d+27))) then
        tmp = (sqrt(x) * y) + 1.0d0
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8e+55) || !(y <= 4.5e+27)) {
		tmp = (Math.sqrt(x) * y) + 1.0;
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8e+55) or not (y <= 4.5e+27):
		tmp = (math.sqrt(x) * y) + 1.0
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8e+55) || !(y <= 4.5e+27))
		tmp = Float64(Float64(sqrt(x) * y) + 1.0);
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8e+55) || ~((y <= 4.5e+27)))
		tmp = (sqrt(x) * y) + 1.0;
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8e+55], N[Not[LessEqual[y, 4.5e+27]], $MachinePrecision]], N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000008e55 or 4.4999999999999999e27 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 93.1%

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

   if -8.00000000000000008e55 < y < 4.4999999999999999e27

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.5%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+55} \lor \neg \left(y \leq 4.5 \cdot 10^{+27}\right):\\ \;\;\;\;\sqrt{x} \cdot y + 1\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+74} \lor \neg \left(y \leq 3.45 \cdot 10^{+96}\right):\\ \;\;\;\;x \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6e+74) (not (<= y 3.45e+96))) (* x (/ y (sqrt x))) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6e+74) || !(y <= 3.45e+96)) {
		tmp = x * (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 <= (-6d+74)) .or. (.not. (y <= 3.45d+96))) then
        tmp = x * (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 <= -6e+74) || !(y <= 3.45e+96)) {
		tmp = x * (y / Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6e+74) or not (y <= 3.45e+96):
		tmp = x * (y / math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6e+74) || !(y <= 3.45e+96))
		tmp = Float64(x * 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 <= -6e+74) || ~((y <= 3.45e+96)))
		tmp = x * (y / sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6e+74], N[Not[LessEqual[y, 3.45e+96]], $MachinePrecision]], N[(x * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6e74 or 3.44999999999999999e96 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 73.0%

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

   4. Taylor expanded in x around 0 70.4%

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

      1. *-commutative 70.4%

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

      2. unpow-1 70.4%

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

      3. metadata-eval 70.4%

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

      4. pow-sqr 70.4%

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

      5. rem-sqrt-square 70.4%

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

      6. rem-square-sqrt 70.2%

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

      7. fabs-sqr 70.2%

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

      8. rem-square-sqrt 70.4%

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

   6. Simplified 70.4%

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

      1. pow1 70.4%

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

      2. associate-*r* 80.9%

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

      3. metadata-eval 80.9%

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

      4. metadata-eval 80.9%

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

      5. sqrt-pow2 80.8%

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

      6. metadata-eval 80.8%

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

      7. inv-pow 80.8%

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

      8. associate-*r* 70.4%

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

      9. *-commutative 70.4%

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

      10. associate-*l/ 70.4%

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

      11. *-un-lft-identity 70.4%

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

   8. Applied egg-rr 70.4%

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

      1. unpow1 70.4%

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

   10. Simplified 70.4%

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

   if -6e74 < y < 3.44999999999999999e96

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 94.2%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+74} \lor \neg \left(y \leq 3.45 \cdot 10^{+96}\right):\\ \;\;\;\;x \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = sqrt(x) * y;
	double tmp;
	if (x <= 1.0) {
		tmp = t_0 + 1.0;
	} 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 = sqrt(x) * y
    if (x <= 1.0d0) then
        tmp = t_0 + 1.0d0
    else
        tmp = t_0 - x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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. Add Preprocessing

   3. Taylor expanded in x around 0 98.4%

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

   if 1 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 99.0%

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

      1. neg-mul-1 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around 0 99.0%

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

      1. mul-1-neg 99.0%

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

      2. +-commutative 99.0%

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

      3. sub-neg 99.0%

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

   8. Simplified 99.0%

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

4. Final simplification 98.7%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 6: 66.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - x \cdot x\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+166}:\\ \;\;\;\;\frac{t\_0}{x + 1}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+145}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x x))))
   (if (<= y -2.9e+166)
     (/ t_0 (+ x 1.0))
     (if (<= y 1.6e+145) (- 1.0 x) (/ t_0 (- 1.0 x))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x * x);
	double tmp;
	if (y <= -2.9e+166) {
		tmp = t_0 / (x + 1.0);
	} else if (y <= 1.6e+145) {
		tmp = 1.0 - x;
	} else {
		tmp = t_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 <= (-2.9d+166)) then
        tmp = t_0 / (x + 1.0d0)
    else if (y <= 1.6d+145) then
        tmp = 1.0d0 - x
    else
        tmp = t_0 / (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 <= -2.9e+166) {
		tmp = t_0 / (x + 1.0);
	} else if (y <= 1.6e+145) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0 / (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x * x)
	tmp = 0
	if y <= -2.9e+166:
		tmp = t_0 / (x + 1.0)
	elif y <= 1.6e+145:
		tmp = 1.0 - x
	else:
		tmp = t_0 / (1.0 - x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x * x))
	tmp = 0.0
	if (y <= -2.9e+166)
		tmp = Float64(t_0 / Float64(x + 1.0));
	elseif (y <= 1.6e+145)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(t_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 <= -2.9e+166)
		tmp = t_0 / (x + 1.0);
	elseif (y <= 1.6e+145)
		tmp = 1.0 - x;
	else
		tmp = t_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, -2.9e+166], N[(t$95$0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+145], N[(1.0 - x), $MachinePrecision], N[(t$95$0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.9000000000000001e166

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 3.4%

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

      1. sub-neg 3.4%

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

      2. flip-+ 22.1%

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

      3. metadata-eval 22.1%

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

   5. Applied egg-rr 22.1%

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

   if -2.9000000000000001e166 < y < 1.60000000000000004e145

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 78.8%

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

   if 1.60000000000000004e145 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 3.7%

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

      1. sub-neg 3.7%

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

      2. flip-+ 3.7%

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

      3. metadata-eval 3.7%

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

   5. Applied egg-rr 3.7%

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

      1. neg-sub0 3.7%

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

      2. sub-neg 3.7%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 4.6%

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

      5. sqr-neg 4.6%

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

      6. sqrt-unprod 25.6%

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

      7. add-sqr-sqrt 25.6%

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

   7. Applied egg-rr 25.6%

      \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \color{blue}{\left(0 + x\right)}} \]
   8. Step-by-step derivation

      1. +-lft-identity 25.6%

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

   9. Simplified 25.6%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+166}:\\ \;\;\;\;\frac{1 - x \cdot x}{x + 1}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+145}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - x \cdot x\\ \mathbf{if}\;y \leq -1.18 \cdot 10^{+171}:\\ \;\;\;\;\frac{t\_0}{x}\\ \mathbf{elif}\;y \leq 10^{+147}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x x))))
   (if (<= y -1.18e+171)
     (/ t_0 x)
     (if (<= y 1e+147) (- 1.0 x) (/ t_0 (- 1.0 x))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x * x);
	double tmp;
	if (y <= -1.18e+171) {
		tmp = t_0 / x;
	} else if (y <= 1e+147) {
		tmp = 1.0 - x;
	} else {
		tmp = t_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 <= (-1.18d+171)) then
        tmp = t_0 / x
    else if (y <= 1d+147) then
        tmp = 1.0d0 - x
    else
        tmp = t_0 / (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 <= -1.18e+171) {
		tmp = t_0 / x;
	} else if (y <= 1e+147) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0 / (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x * x)
	tmp = 0
	if y <= -1.18e+171:
		tmp = t_0 / x
	elif y <= 1e+147:
		tmp = 1.0 - x
	else:
		tmp = t_0 / (1.0 - x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x * x))
	tmp = 0.0
	if (y <= -1.18e+171)
		tmp = Float64(t_0 / x);
	elseif (y <= 1e+147)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(t_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 <= -1.18e+171)
		tmp = t_0 / x;
	elseif (y <= 1e+147)
		tmp = 1.0 - x;
	else
		tmp = t_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, -1.18e+171], N[(t$95$0 / x), $MachinePrecision], If[LessEqual[y, 1e+147], N[(1.0 - x), $MachinePrecision], N[(t$95$0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1799999999999999e171

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 3.3%

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

      1. sub-neg 3.3%

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

      2. flip-+ 23.3%

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

      3. metadata-eval 23.3%

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

   5. Applied egg-rr 23.3%

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

   6. Taylor expanded in x around inf 23.1%

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

   if -1.1799999999999999e171 < y < 9.9999999999999998e146

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 78.0%

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

   if 9.9999999999999998e146 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 3.7%

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

      1. sub-neg 3.7%

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

      2. flip-+ 3.7%

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

      3. metadata-eval 3.7%

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

   5. Applied egg-rr 3.7%

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

      1. neg-sub0 3.7%

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

      2. sub-neg 3.7%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 4.6%

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

      5. sqr-neg 4.6%

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

      6. sqrt-unprod 25.6%

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

      7. add-sqr-sqrt 25.6%

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

   7. Applied egg-rr 25.6%

      \[\leadsto \frac{1 - \left(-x\right) \cdot \left(-x\right)}{1 - \color{blue}{\left(0 + x\right)}} \]
   8. Step-by-step derivation

      1. +-lft-identity 25.6%

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

   9. Simplified 25.6%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{+171}:\\ \;\;\;\;\frac{1 - x \cdot x}{x}\\ \mathbf{elif}\;y \leq 10^{+147}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+170}:\\ \;\;\;\;\frac{1 - x \cdot x}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.4e+170) (/ (- 1.0 (* x x)) x) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.4e+170) {
		tmp = (1.0 - (x * x)) / 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 <= (-7.4d+170)) then
        tmp = (1.0d0 - (x * x)) / 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 <= -7.4e+170) {
		tmp = (1.0 - (x * x)) / x;
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.4e+170:
		tmp = (1.0 - (x * x)) / x
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.4e+170)
		tmp = Float64(Float64(1.0 - Float64(x * x)) / x);
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.4e+170)
		tmp = (1.0 - (x * x)) / x;
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.4e+170], N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.39999999999999975e170

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 3.3%

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

      1. sub-neg 3.3%

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

      2. flip-+ 23.3%

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

      3. metadata-eval 23.3%

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

   5. Applied egg-rr 23.3%

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

   6. Taylor expanded in x around inf 23.1%

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

   if -7.39999999999999975e170 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 66.6%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+170}:\\ \;\;\;\;\frac{1 - x \cdot x}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.1% accurate, 15.3× 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. Add Preprocessing

   3. Taylor expanded in y around 0 61.1%

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

   4. Taylor expanded in x around 0 59.9%

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

   if 1 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 58.2%

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

   4. Taylor expanded in x around inf 57.3%

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

      1. neg-mul-1 99.0%

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

   6. Simplified 57.3%

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

Alternative 10: 62.1% 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 99.9%

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

3. Taylor expanded in y around 0 59.7%

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

Alternative 11: 31.2% 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. Add Preprocessing

3. Taylor expanded in y around 0 59.7%

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

4. Taylor expanded in x around 0 30.9%

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

Reproduce

herbie shell --seed 2024135 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))