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

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))

C

double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + ((0.99229d0 + (x * 0.04481d0)) * x)))
end function

Java

public static double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
}

Python

def code(x):
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)))

Julia

function code(x)
	return Float64(x - Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(Float64(0.99229 + Float64(x * 0.04481)) * x))))
end

MATLAB

function tmp = code(x)
	tmp = x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
end

Wolfram

code[x_] := N[(x - N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))

C

double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + ((0.99229d0 + (x * 0.04481d0)) * x)))
end function

Java

public static double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
}

Python

def code(x):
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)))

Julia

function code(x)
	return Float64(x - Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(Float64(0.99229 + Float64(x * 0.04481)) * x))))
end

MATLAB

function tmp = code(x)
	tmp = x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
end

Wolfram

code[x_] := N[(x - N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ x + \frac{2.30753 + x \cdot 0.27061}{-1 + \frac{x \cdot \left({x}^{2} \cdot 0.0020079361 - 0.9846394441\right)}{0.99229 + x \cdot -0.04481}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  x
  (/
   (+ 2.30753 (* x 0.27061))
   (+
    -1.0
    (/
     (* x (- (* (pow x 2.0) 0.0020079361) 0.9846394441))
     (+ 0.99229 (* x -0.04481)))))))

C

double code(double x) {
	return x + ((2.30753 + (x * 0.27061)) / (-1.0 + ((x * ((pow(x, 2.0) * 0.0020079361) - 0.9846394441)) / (0.99229 + (x * -0.04481)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((2.30753d0 + (x * 0.27061d0)) / ((-1.0d0) + ((x * (((x ** 2.0d0) * 0.0020079361d0) - 0.9846394441d0)) / (0.99229d0 + (x * (-0.04481d0))))))
end function

Java

public static double code(double x) {
	return x + ((2.30753 + (x * 0.27061)) / (-1.0 + ((x * ((Math.pow(x, 2.0) * 0.0020079361) - 0.9846394441)) / (0.99229 + (x * -0.04481)))));
}

Python

def code(x):
	return x + ((2.30753 + (x * 0.27061)) / (-1.0 + ((x * ((math.pow(x, 2.0) * 0.0020079361) - 0.9846394441)) / (0.99229 + (x * -0.04481)))))

Julia

function code(x)
	return Float64(x + Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(-1.0 + Float64(Float64(x * Float64(Float64((x ^ 2.0) * 0.0020079361) - 0.9846394441)) / Float64(0.99229 + Float64(x * -0.04481))))))
end

MATLAB

function tmp = code(x)
	tmp = x + ((2.30753 + (x * 0.27061)) / (-1.0 + ((x * (((x ^ 2.0) * 0.0020079361) - 0.9846394441)) / (0.99229 + (x * -0.04481)))));
end

Wolfram

code[x_] := N[(x + N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(N[(x * N[(N[(N[Power[x, 2.0], $MachinePrecision] * 0.0020079361), $MachinePrecision] - 0.9846394441), $MachinePrecision]), $MachinePrecision] / N[(0.99229 + N[(x * -0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-+ 100.0%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\frac{0.99229 \cdot 0.99229 - \left(x \cdot 0.04481\right) \cdot \left(x \cdot 0.04481\right)}{0.99229 - x \cdot 0.04481}} \cdot x} \]

   2. associate-*l/ 100.0%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\frac{\left(0.99229 \cdot 0.99229 - \left(x \cdot 0.04481\right) \cdot \left(x \cdot 0.04481\right)\right) \cdot x}{0.99229 - x \cdot 0.04481}}} \]

   3. metadata-eval 100.0%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(\color{blue}{0.9846394441} - \left(x \cdot 0.04481\right) \cdot \left(x \cdot 0.04481\right)\right) \cdot x}{0.99229 - x \cdot 0.04481}} \]

   4. swap-sqr 100.0%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(0.9846394441 - \color{blue}{\left(x \cdot x\right) \cdot \left(0.04481 \cdot 0.04481\right)}\right) \cdot x}{0.99229 - x \cdot 0.04481}} \]

   5. pow2 100.0%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(0.9846394441 - \color{blue}{{x}^{2}} \cdot \left(0.04481 \cdot 0.04481\right)\right) \cdot x}{0.99229 - x \cdot 0.04481}} \]

   6. metadata-eval 100.0%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(0.9846394441 - {x}^{2} \cdot \color{blue}{0.0020079361}\right) \cdot x}{0.99229 - x \cdot 0.04481}} \]

   7. *-commutative 100.0%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(0.9846394441 - {x}^{2} \cdot 0.0020079361\right) \cdot x}{0.99229 - \color{blue}{0.04481 \cdot x}}} \]

   8. cancel-sign-sub-inv 100.0%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(0.9846394441 - {x}^{2} \cdot 0.0020079361\right) \cdot x}{\color{blue}{0.99229 + \left(-0.04481\right) \cdot x}}} \]

   9. metadata-eval 100.0%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(0.9846394441 - {x}^{2} \cdot 0.0020079361\right) \cdot x}{0.99229 + \color{blue}{-0.04481} \cdot x}} \]

4. Applied egg-rr 100.0%

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\frac{\left(0.9846394441 - {x}^{2} \cdot 0.0020079361\right) \cdot x}{0.99229 + -0.04481 \cdot x}}} \]

5. Final simplification 100.0%

   \[\leadsto x + \frac{2.30753 + x \cdot 0.27061}{-1 + \frac{x \cdot \left({x}^{2} \cdot 0.0020079361 - 0.9846394441\right)}{0.99229 + x \cdot -0.04481}} \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{2.30753 + x \cdot 0.27061}{-1 - x \cdot \left(0.99229 + x \cdot 0.04481\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ x (/ (+ 2.30753 (* x 0.27061)) (- -1.0 (* x (+ 0.99229 (* x 0.04481)))))))

C

double code(double x) {
	return x + ((2.30753 + (x * 0.27061)) / (-1.0 - (x * (0.99229 + (x * 0.04481)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((2.30753d0 + (x * 0.27061d0)) / ((-1.0d0) - (x * (0.99229d0 + (x * 0.04481d0)))))
end function

Java

public static double code(double x) {
	return x + ((2.30753 + (x * 0.27061)) / (-1.0 - (x * (0.99229 + (x * 0.04481)))));
}

Python

def code(x):
	return x + ((2.30753 + (x * 0.27061)) / (-1.0 - (x * (0.99229 + (x * 0.04481)))))

Julia

function code(x)
	return Float64(x + Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(-1.0 - Float64(x * Float64(0.99229 + Float64(x * 0.04481))))))
end

MATLAB

function tmp = code(x)
	tmp = x + ((2.30753 + (x * 0.27061)) / (-1.0 - (x * (0.99229 + (x * 0.04481)))));
end

Wolfram

code[x_] := N[(x + N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto x + \frac{2.30753 + x \cdot 0.27061}{-1 - x \cdot \left(0.99229 + x \cdot 0.04481\right)} \]
4. Add Preprocessing

Alternative 3: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x - \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot 0.99229} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x 0.99229)))))

C

double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + (x * 0.99229)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * 0.99229d0)))
end function

Java

public static double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + (x * 0.99229)));
}

Python

def code(x):
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + (x * 0.99229)))

Julia

function code(x)
	return Float64(x - Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * 0.99229))))
end

MATLAB

function tmp = code(x)
	tmp = x - ((2.30753 + (x * 0.27061)) / (1.0 + (x * 0.99229)));
end

Wolfram

code[x_] := N[(x - N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * 0.99229), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.1%

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{0.99229 \cdot x}} \]
4. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} \]

5. Simplified 98.1%

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} \]

6. Final simplification 98.1%

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot 0.99229} \]
7. Add Preprocessing

Alternative 4: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05) x (if (<= x 1.15) -2.30753 x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x;
	} else if (x <= 1.15) {
		tmp = -2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = x
    else if (x <= 1.15d0) then
        tmp = -2.30753d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x;
	} else if (x <= 1.15) {
		tmp = -2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = x
	elif x <= 1.15:
		tmp = -2.30753
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = x;
	elseif (x <= 1.15)
		tmp = -2.30753;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = x;
	elseif (x <= 1.15)
		tmp = -2.30753;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], x, If[LessEqual[x, 1.15], -2.30753, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < x

   1. Initial program 100.0%

      \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.5%

      \[\leadsto x - \color{blue}{2.30753} \]

   4. Taylor expanded in x around inf 99.4%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   1. Initial program 99.9%

      \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 96.5%

      \[\leadsto x - \color{blue}{2.30753} \]

   4. Taylor expanded in x around 0 96.5%

      \[\leadsto \color{blue}{-2.30753} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ x - 2.30753 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- x 2.30753))

C

double code(double x) {
	return x - 2.30753;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - 2.30753d0
end function

Java

public static double code(double x) {
	return x - 2.30753;
}

Python

def code(x):
	return x - 2.30753

Julia

function code(x)
	return Float64(x - 2.30753)
end

MATLAB

function tmp = code(x)
	tmp = x - 2.30753;
end

Wolfram

code[x_] := N[(x - 2.30753), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 97.5%

   \[\leadsto x - \color{blue}{2.30753} \]

4. Final simplification 97.5%

   \[\leadsto x - 2.30753 \]
5. Add Preprocessing

Alternative 6: 50.5% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ -2.30753 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.30753)

C

double code(double x) {
	return -2.30753;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -2.30753d0
end function

Java

public static double code(double x) {
	return -2.30753;
}

Python

def code(x):
	return -2.30753

Julia

function code(x)
	return -2.30753
end

MATLAB

function tmp = code(x)
	tmp = -2.30753;
end

Wolfram

code[x_] := -2.30753

Derivation

1. Initial program 100.0%

   \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 97.5%

   \[\leadsto x - \color{blue}{2.30753} \]

4. Taylor expanded in x around 0 52.8%

   \[\leadsto \color{blue}{-2.30753} \]

5. Final simplification 52.8%

   \[\leadsto -2.30753 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))