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

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

Alternatives: 9

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

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

FPCore

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

C

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

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)))) * (((x ** 2.0d0) * 0.0020079361d0) - 0.9846394441d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x + N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(N[(x / N[(0.99229 + N[(x * -0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * 0.0020079361), $MachinePrecision] - 0.9846394441), $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. +-commutative 100.0%

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

   2. fma-undefine 100.0%

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

   3. *-commutative 100.0%

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

   4. fma-undefine 100.0%

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

   5. +-commutative 100.0%

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

   6. flip-+ 100.0%

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

   7. associate-*r/ 100.0%

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

   8. metadata-eval 100.0%

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

   9. swap-sqr 100.0%

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

   10. pow2 100.0%

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

   11. metadata-eval 100.0%

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

   12. *-commutative 100.0%

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

   13. cancel-sign-sub-inv 100.0%

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

   14. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-/l* 100.0%

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

   3. *-commutative 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9:\\ \;\;\;\;x + \frac{\frac{82.23527511657367}{x} - 6.039053782637804}{x}\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;x \cdot \left(3.0191289437 + x \cdot -1.900161040244073\right) - 2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.9)
   (+ x (/ (- (/ 82.23527511657367 x) 6.039053782637804) x))
   (if (<= x 1.2)
     (- (* x (+ 3.0191289437 (* x -1.900161040244073))) 2.30753)
     x)))

C

double code(double x) {
	double tmp;
	if (x <= -4.9) {
		tmp = x + (((82.23527511657367 / x) - 6.039053782637804) / x);
	} else if (x <= 1.2) {
		tmp = (x * (3.0191289437 + (x * -1.900161040244073))) - 2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.9d0)) then
        tmp = x + (((82.23527511657367d0 / x) - 6.039053782637804d0) / x)
    else if (x <= 1.2d0) then
        tmp = (x * (3.0191289437d0 + (x * (-1.900161040244073d0)))) - 2.30753d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.9) {
		tmp = x + (((82.23527511657367 / x) - 6.039053782637804) / x);
	} else if (x <= 1.2) {
		tmp = (x * (3.0191289437 + (x * -1.900161040244073))) - 2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.9:
		tmp = x + (((82.23527511657367 / x) - 6.039053782637804) / x)
	elif x <= 1.2:
		tmp = (x * (3.0191289437 + (x * -1.900161040244073))) - 2.30753
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.9)
		tmp = Float64(x + Float64(Float64(Float64(82.23527511657367 / x) - 6.039053782637804) / x));
	elseif (x <= 1.2)
		tmp = Float64(Float64(x * Float64(3.0191289437 + Float64(x * -1.900161040244073))) - 2.30753);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.9)
		tmp = x + (((82.23527511657367 / x) - 6.039053782637804) / x);
	elseif (x <= 1.2)
		tmp = (x * (3.0191289437 + (x * -1.900161040244073))) - 2.30753;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.9], N[(x + N[(N[(N[(82.23527511657367 / x), $MachinePrecision] - 6.039053782637804), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2], N[(N[(x * N[(3.0191289437 + N[(x * -1.900161040244073), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.30753), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9000000000000004

   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 inf 98.2%

      \[\leadsto x - \color{blue}{\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ 98.2%

         \[\leadsto x - \frac{6.039053782637804 - \color{blue}{\frac{82.23527511657367 \cdot 1}{x}}}{x} \]

      2. metadata-eval 98.2%

         \[\leadsto x - \frac{6.039053782637804 - \frac{\color{blue}{82.23527511657367}}{x}}{x} \]

   5. Simplified 98.2%

      \[\leadsto x - \color{blue}{\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x}} \]

   if -4.9000000000000004 < x < 1.19999999999999996

   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 98.7%

      \[\leadsto x - \color{blue}{\left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 2.0191289437\right)\right)} \]

   4. Taylor expanded in x around 0 98.7%

      \[\leadsto \color{blue}{x \cdot \left(3.0191289437 + -1.900161040244073 \cdot x\right) - 2.30753} \]

   if 1.19999999999999996 < 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 99.1%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9:\\ \;\;\;\;x + \frac{\frac{82.23527511657367}{x} - 6.039053782637804}{x}\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;x \cdot \left(3.0191289437 + x \cdot -1.900161040244073\right) - 2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;x - \frac{6.039053782637804}{x}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot 3.0191289437 - 2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.5)
   (- x (/ 6.039053782637804 x))
   (if (<= x 1.15) (- (* x 3.0191289437) 2.30753) x)))

C

double code(double x) {
	double tmp;
	if (x <= -2.5) {
		tmp = x - (6.039053782637804 / x);
	} else if (x <= 1.15) {
		tmp = (x * 3.0191289437) - 2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.5d0)) then
        tmp = x - (6.039053782637804d0 / x)
    else if (x <= 1.15d0) then
        tmp = (x * 3.0191289437d0) - 2.30753d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.5) {
		tmp = x - (6.039053782637804 / x);
	} else if (x <= 1.15) {
		tmp = (x * 3.0191289437) - 2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.5:
		tmp = x - (6.039053782637804 / x)
	elif x <= 1.15:
		tmp = (x * 3.0191289437) - 2.30753
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.5)
		tmp = Float64(x - Float64(6.039053782637804 / x));
	elseif (x <= 1.15)
		tmp = Float64(Float64(x * 3.0191289437) - 2.30753);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.5)
		tmp = x - (6.039053782637804 / x);
	elseif (x <= 1.15)
		tmp = (x * 3.0191289437) - 2.30753;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.5], N[(x - N[(6.039053782637804 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15], N[(N[(x * 3.0191289437), $MachinePrecision] - 2.30753), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5

   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 inf 98.0%

      \[\leadsto x - \color{blue}{\frac{6.039053782637804}{x}} \]

   if -2.5 < 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 97.8%

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

   4. Taylor expanded in x around 0 97.8%

      \[\leadsto \color{blue}{3.0191289437 \cdot x - 2.30753} \]

   if 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 99.1%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;x - \frac{6.039053782637804}{x}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot 3.0191289437 - 2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4:\\ \;\;\;\;x + \frac{\frac{82.23527511657367}{x} - 6.039053782637804}{x}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot 3.0191289437 - 2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.4)
   (+ x (/ (- (/ 82.23527511657367 x) 6.039053782637804) x))
   (if (<= x 1.15) (- (* x 3.0191289437) 2.30753) x)))

C

double code(double x) {
	double tmp;
	if (x <= -5.4) {
		tmp = x + (((82.23527511657367 / x) - 6.039053782637804) / x);
	} else if (x <= 1.15) {
		tmp = (x * 3.0191289437) - 2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5.4d0)) then
        tmp = x + (((82.23527511657367d0 / x) - 6.039053782637804d0) / x)
    else if (x <= 1.15d0) then
        tmp = (x * 3.0191289437d0) - 2.30753d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.4) {
		tmp = x + (((82.23527511657367 / x) - 6.039053782637804) / x);
	} else if (x <= 1.15) {
		tmp = (x * 3.0191289437) - 2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.4:
		tmp = x + (((82.23527511657367 / x) - 6.039053782637804) / x)
	elif x <= 1.15:
		tmp = (x * 3.0191289437) - 2.30753
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.4)
		tmp = Float64(x + Float64(Float64(Float64(82.23527511657367 / x) - 6.039053782637804) / x));
	elseif (x <= 1.15)
		tmp = Float64(Float64(x * 3.0191289437) - 2.30753);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.4)
		tmp = x + (((82.23527511657367 / x) - 6.039053782637804) / x);
	elseif (x <= 1.15)
		tmp = (x * 3.0191289437) - 2.30753;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.4], N[(x + N[(N[(N[(82.23527511657367 / x), $MachinePrecision] - 6.039053782637804), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15], N[(N[(x * 3.0191289437), $MachinePrecision] - 2.30753), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if x < -5.4000000000000004

   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 inf 98.2%

      \[\leadsto x - \color{blue}{\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ 98.2%

         \[\leadsto x - \frac{6.039053782637804 - \color{blue}{\frac{82.23527511657367 \cdot 1}{x}}}{x} \]

      2. metadata-eval 98.2%

         \[\leadsto x - \frac{6.039053782637804 - \frac{\color{blue}{82.23527511657367}}{x}}{x} \]

   5. Simplified 98.2%

      \[\leadsto x - \color{blue}{\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x}} \]

   if -5.4000000000000004 < 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 97.8%

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

   4. Taylor expanded in x around 0 97.8%

      \[\leadsto \color{blue}{3.0191289437 \cdot x - 2.30753} \]

   if 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 99.1%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4:\\ \;\;\;\;x + \frac{\frac{82.23527511657367}{x} - 6.039053782637804}{x}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot 3.0191289437 - 2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% 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 7: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -1.05) x (if (<= x 1.2) -2.30753 x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x;
	} else if (x <= 1.2) {
		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 <= 1.2d0) 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 <= 1.2) {
		tmp = -2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = x
	elif x <= 1.2:
		tmp = -2.30753
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = x;
	elseif (x <= 1.2)
		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 <= 1.2)
		tmp = -2.30753;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], x, If[LessEqual[x, 1.2], -2.30753, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.19999999999999996 < 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 97.8%

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

   4. Taylor expanded in x around inf 98.9%

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

   if -1.05000000000000004 < x < 1.19999999999999996

   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 95.5%

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

   4. Taylor expanded in x around 0 95.4%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 96.5%

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

4. Final simplification 96.5%

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

Alternative 9: 50.1% 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 96.5%

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

4. Taylor expanded in x around 0 53.7%

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

5. Final simplification 53.7%

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

Reproduce

herbie shell --seed 2024115 
(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)))))