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

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x - N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-1.0 + N[(N[(x * 0.99229), $MachinePrecision] + N[(1.0 + N[(0.04481 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. expm1-log1p-u 100.0%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.99229 + x \cdot 0.04481\right) \cdot x\right)\right)}} \]

   2. expm1-udef 100.0%

      \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(0.99229 + x \cdot 0.04481\right) \cdot x\right)} - 1\right)}} \]

   3. log1p-udef 100.0%

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

   4. add-exp-log 100.0%

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

   5. flip-+ 77.7%

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

   6. div-inv 77.7%

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

   7. fma-neg 77.7%

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

3. Applied egg-rr 77.7%

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\mathsf{fma}\left(1 - {\left(x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)\right)}^{2}, \frac{1}{1 - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}, -1\right)}} \]
4. Step-by-step derivation

   1. fma-udef 77.7%

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

   2. +-commutative 77.7%

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

   3. associate-*r/ 77.7%

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

   4. *-rgt-identity 77.7%

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

5. Simplified 77.7%

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

6. Taylor expanded in x around 0 100.0%

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

7. Final simplification 100.0%

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

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. Final simplification 100.0%

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

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. Taylor expanded in x around 0 99.3%

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

   1. *-commutative 99.3%

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

4. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 4: 98.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -3.6) x (if (<= x 1.15) -2.30753 x)))

C

double code(double x) {
	double tmp;
	if (x <= -3.6) {
		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 <= (-3.6d0)) 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 <= -3.6) {
		tmp = x;
	} else if (x <= 1.15) {
		tmp = -2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.6:
		tmp = x
	elif x <= 1.15:
		tmp = -2.30753
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.6)
		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 <= -3.6)
		tmp = x;
	elseif (x <= 1.15)
		tmp = -2.30753;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.6], x, If[LessEqual[x, 1.15], -2.30753, x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000009 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. Taylor expanded in x around 0 98.8%

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

   3. Taylor expanded in x around inf 99.5%

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

   if -3.60000000000000009 < x < 1.1499999999999999

   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. Taylor expanded in x around 0 98.9%

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

   3. Taylor expanded in x around 0 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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. Taylor expanded in x around 0 98.8%

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

3. Final simplification 98.8%

   \[\leadsto x - 2.30753 \]

Alternative 6: 50.6% 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. Taylor expanded in x around 0 98.8%

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

3. Taylor expanded in x around 0 52.7%

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

4. Final simplification 52.7%

   \[\leadsto -2.30753 \]

Reproduce

herbie shell --seed 2023195 
(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)))))