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

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 5

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, 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 2: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.65:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.65) x (if (<= x 1.15) -2.30753 x)))

C

double code(double x) {
	double tmp;
	if (x <= -3.65) {
		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 <= (-3.65d0)) 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 <= -3.65) {
		tmp = x;
	} else if (x <= 1.15) {
		tmp = -2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.65:
		tmp = x
	elif x <= 1.15:
		tmp = -2.30753
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.65)
		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 <= -3.65)
		tmp = x;
	elseif (x <= 1.15)
		tmp = -2.30753;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.65], x, If[LessEqual[x, 1.15], -2.30753, x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.64999999999999991 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 99.7%

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

   4. Taylor expanded in x around inf 99.7%

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

   if -3.64999999999999991 < 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 97.9%

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

   4. Taylor expanded in x around 0 97.4%

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

Alternative 3: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ x + \frac{-5.3246947009}{2.30753 - x \cdot -2.0191289437} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ x (/ -5.3246947009 (- 2.30753 (* x -2.0191289437)))))

C

double code(double x) {
	return x + (-5.3246947009 / (2.30753 - (x * -2.0191289437)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((-5.3246947009d0) / (2.30753d0 - (x * (-2.0191289437d0))))
end function

Java

public static double code(double x) {
	return x + (-5.3246947009 / (2.30753 - (x * -2.0191289437)));
}

Python

def code(x):
	return x + (-5.3246947009 / (2.30753 - (x * -2.0191289437)))

Julia

function code(x)
	return Float64(x + Float64(-5.3246947009 / Float64(2.30753 - Float64(x * -2.0191289437))))
end

MATLAB

function tmp = code(x)
	tmp = x + (-5.3246947009 / (2.30753 - (x * -2.0191289437)));
end

Wolfram

code[x_] := N[(x + N[(-5.3246947009 / N[(2.30753 - N[(x * -2.0191289437), $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 53.8%

   \[\leadsto x - \color{blue}{\left(2.30753 + -2.0191289437 \cdot x\right)} \]
4. Step-by-step derivation

   1. *-commutative 53.8%

      \[\leadsto x - \left(2.30753 + \color{blue}{x \cdot -2.0191289437}\right) \]

5. Simplified 53.8%

   \[\leadsto x - \color{blue}{\left(2.30753 + x \cdot -2.0191289437\right)} \]
6. Step-by-step derivation

   1. +-commutative 53.8%

      \[\leadsto x - \color{blue}{\left(x \cdot -2.0191289437 + 2.30753\right)} \]

   2. flip-+ 50.6%

      \[\leadsto x - \color{blue}{\frac{\left(x \cdot -2.0191289437\right) \cdot \left(x \cdot -2.0191289437\right) - 2.30753 \cdot 2.30753}{x \cdot -2.0191289437 - 2.30753}} \]

   3. swap-sqr 50.6%

      \[\leadsto x - \frac{\color{blue}{\left(x \cdot x\right) \cdot \left(-2.0191289437 \cdot -2.0191289437\right)} - 2.30753 \cdot 2.30753}{x \cdot -2.0191289437 - 2.30753} \]

   4. pow2 50.6%

      \[\leadsto x - \frac{\color{blue}{{x}^{2}} \cdot \left(-2.0191289437 \cdot -2.0191289437\right) - 2.30753 \cdot 2.30753}{x \cdot -2.0191289437 - 2.30753} \]

   5. metadata-eval 50.6%

      \[\leadsto x - \frac{{x}^{2} \cdot \color{blue}{4.076881691287078} - 2.30753 \cdot 2.30753}{x \cdot -2.0191289437 - 2.30753} \]

   6. metadata-eval 49.9%

      \[\leadsto x - \frac{{x}^{2} \cdot 4.076881691287078 - \color{blue}{5.3246947009}}{x \cdot -2.0191289437 - 2.30753} \]

7. Applied egg-rr 49.9%

   \[\leadsto x - \color{blue}{\frac{{x}^{2} \cdot 4.076881691287078 - 5.3246947009}{x \cdot -2.0191289437 - 2.30753}} \]

8. Taylor expanded in x around 0 99.0%

   \[\leadsto x - \frac{\color{blue}{-5.3246947009}}{x \cdot -2.0191289437 - 2.30753} \]

9. Final simplification 99.0%

   \[\leadsto x + \frac{-5.3246947009}{2.30753 - x \cdot -2.0191289437} \]
10. Add Preprocessing

Alternative 4: 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 53.8%

   \[\leadsto x - \color{blue}{\left(2.30753 + -2.0191289437 \cdot x\right)} \]
4. Step-by-step derivation

   1. *-commutative 53.8%

      \[\leadsto x - \left(2.30753 + \color{blue}{x \cdot -2.0191289437}\right) \]

5. Simplified 53.8%

   \[\leadsto x - \color{blue}{\left(2.30753 + x \cdot -2.0191289437\right)} \]

6. Taylor expanded in x around 0 98.4%

   \[\leadsto x - \color{blue}{2.30753} \]
7. Add Preprocessing

Alternative 5: 50.7% 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 98.9%

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

4. Taylor expanded in x around 0 44.7%

   \[\leadsto \color{blue}{-2.30753} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024181 
(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)))))