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

Percentage Accurate: 99.9% → 99.9%

Time: 7.3s

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+70} \lor \neg \left(y \leq 1100000000000\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.8e+70) (not (<= y 1100000000000.0)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.8e+70) || !(y <= 1100000000000.0)) {
		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 <= (-5.8d+70)) .or. (.not. (y <= 1100000000000.0d0))) 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 <= -5.8e+70) || !(y <= 1100000000000.0)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.8e+70) or not (y <= 1100000000000.0):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.8e+70) || !(y <= 1100000000000.0))
		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 <= -5.8e+70) || ~((y <= 1100000000000.0)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.8e+70], N[Not[LessEqual[y, 1100000000000.0]], $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 < -5.7999999999999997e70 or 1.1e12 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 97.0%

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

   if -5.7999999999999997e70 < y < 1.1e12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.8%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+70} \lor \neg \left(y \leq 1100000000000\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+70} \lor \neg \left(y \leq 4.8 \cdot 10^{+82}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.8e+70) (not (<= y 4.8e+82))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.8e+70) || !(y <= 4.8e+82)) {
		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 <= (-5.8d+70)) .or. (.not. (y <= 4.8d+82))) 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 <= -5.8e+70) || !(y <= 4.8e+82)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.8e+70) or not (y <= 4.8e+82):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.8e+70) || !(y <= 4.8e+82))
		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 <= -5.8e+70) || ~((y <= 4.8e+82)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.8e+70], N[Not[LessEqual[y, 4.8e+82]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.7999999999999997e70 or 4.79999999999999996e82 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. div-sub 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0 98.4%

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

   7. Taylor expanded in x around inf 94.0%

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

   if -5.7999999999999997e70 < y < 4.79999999999999996e82

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 96.6%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+70} \lor \neg \left(y \leq 4.8 \cdot 10^{+82}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   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 98.2%

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

      1. neg-mul-1 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. +-commutative 98.2%

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

      3. sub-neg 98.2%

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

   8. Simplified 98.2%

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

4. Final simplification 98.3%

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

Alternative 5: 61.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10200:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 10200.0) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 10200.0) {
		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 <= 10200.0d0) 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 <= 10200.0) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 10200.0:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 10200.0)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 10200.0)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 10200.0], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 10200

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 67.0%

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

   4. Taylor expanded in x around 0 65.6%

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

   if 10200 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.1%

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

      1. neg-mul-1 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in y around 0 65.0%

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

      1. mul-1-neg 65.0%

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

   8. Simplified 65.0%

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

Alternative 6: 64.1% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf 88.6%

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

   1. associate--l+ 88.6%

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

   2. div-sub 88.6%

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

5. Simplified 88.6%

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

6. Taylor expanded in y around 0 55.5%

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

   1. associate-*r/ 68.1%

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

8. Applied egg-rr 68.1%

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

9. Final simplification 68.1%

   \[\leadsto \frac{\left(1 - x\right) \cdot y}{y} \]
10. Add Preprocessing

Alternative 7: 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 99.9%

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

3. Taylor expanded in y around 0 66.9%

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

Alternative 8: 31.4% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0 66.9%

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

4. Taylor expanded in x around 0 36.5%

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

Reproduce

herbie shell --seed 2024116 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))