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

Percentage Accurate: 99.9% → 99.9%

Time: 6.5s

Alternatives: 7

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 94.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+72} \lor \neg \left(y \leq 4.5 \cdot 10^{+74}\right):\\ \;\;\;\;1 + \sqrt{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.8e+72) (not (<= y 4.5e+74)))
   (+ 1.0 (* (sqrt x) y))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.8e+72) || !(y <= 4.5e+74)) {
		tmp = 1.0 + (sqrt(x) * y);
	} 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 <= (-2.8d+72)) .or. (.not. (y <= 4.5d+74))) then
        tmp = 1.0d0 + (sqrt(x) * y)
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.8e+72) || !(y <= 4.5e+74)) {
		tmp = 1.0 + (Math.sqrt(x) * y);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.8e+72) or not (y <= 4.5e+74):
		tmp = 1.0 + (math.sqrt(x) * y)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.8e+72) || !(y <= 4.5e+74))
		tmp = Float64(1.0 + Float64(sqrt(x) * y));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.8e+72) || ~((y <= 4.5e+74)))
		tmp = 1.0 + (sqrt(x) * y);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.8e+72], N[Not[LessEqual[y, 4.5e+74]], $MachinePrecision]], N[(1.0 + N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7999999999999999e72 or 4.5e74 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 95.0%

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

   if -2.7999999999999999e72 < y < 4.5e74

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 95.3%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+72} \lor \neg \left(y \leq 4.5 \cdot 10^{+74}\right):\\ \;\;\;\;1 + \sqrt{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+82} \lor \neg \left(y \leq 4.4 \cdot 10^{+74}\right):\\ \;\;\;\;\sqrt{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.7e+82) (not (<= y 4.4e+74))) (* (sqrt x) y) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.7e+82) || !(y <= 4.4e+74)) {
		tmp = sqrt(x) * y;
	} 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 <= (-5.7d+82)) .or. (.not. (y <= 4.4d+74))) then
        tmp = sqrt(x) * y
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5.7e+82) || !(y <= 4.4e+74)) {
		tmp = Math.sqrt(x) * y;
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.7e+82) or not (y <= 4.4e+74):
		tmp = math.sqrt(x) * y
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.7e+82) || !(y <= 4.4e+74))
		tmp = Float64(sqrt(x) * y);
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5.7e+82) || ~((y <= 4.4e+74)))
		tmp = sqrt(x) * y;
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.7e+82], N[Not[LessEqual[y, 4.4e+74]], $MachinePrecision]], N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.70000000000000016e82 or 4.4000000000000002e74 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 68.5%

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

   4. Taylor expanded in y around inf 94.0%

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

      1. mul-1-neg 94.0%

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

      2. unsub-neg 94.0%

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

   6. Simplified 94.0%

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

   7. Taylor expanded in x around 0 89.1%

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

   if -5.70000000000000016e82 < y < 4.4000000000000002e74

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 94.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+82} \lor \neg \left(y \leq 4.4 \cdot 10^{+74}\right):\\ \;\;\;\;\sqrt{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = sqrt(x) * y;
	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 = sqrt(x) * y
    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 = Math.sqrt(x) * y;
	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 = math.sqrt(x) * y
	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(sqrt(x) * y)
	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 = sqrt(x) * y;
	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[(N[Sqrt[x], $MachinePrecision] * y), $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 98.2%

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

   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 99.2%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

      2. +-commutative 99.3%

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

      3. sub-neg 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 98.7%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 6: 62.7% 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 62.7%

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

4. Final simplification 62.7%

   \[\leadsto 1 - x \]
5. Add Preprocessing

Alternative 7: 32.7% accurate, 53.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf 57.8%

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

4. Taylor expanded in y around 0 30.9%

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

   1. neg-mul-1 30.9%

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

6. Simplified 30.9%

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

7. Final simplification 30.9%

   \[\leadsto -x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024078 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))