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

Percentage Accurate: 100.0% → 100.0%

Time: 8.8s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x + N[(-1.0 / N[(N[(x * N[(N[(x * 0.04481), $MachinePrecision] + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $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. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. +-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. fma-define 100.0%

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

   6. +-commutative 100.0%

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

   7. fma-define 100.0%

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

   8. +-commutative 100.0%

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

   9. fma-define 100.0%

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

4. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

6. Applied egg-rr 100.0%

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

   1. fma-undefine 100.0%

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

8. Applied egg-rr 100.0%

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

   1. fma-undefine 100.0%

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

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 3: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x + \frac{-1}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + x \cdot -0.025050834237766436\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  x
  (/
   -1.0
   (+
    0.4333638132548656
    (* x (+ 0.37920088514346545 (* x -0.025050834237766436)))))))

C

double code(double x) {
	return x + (-1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((-1.0d0) / (0.4333638132548656d0 + (x * (0.37920088514346545d0 + (x * (-0.025050834237766436d0))))))
end function

Java

public static double code(double x) {
	return x + (-1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436)))));
}

Python

def code(x):
	return x + (-1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436)))))

Julia

function code(x)
	return Float64(x + Float64(-1.0 / Float64(0.4333638132548656 + Float64(x * Float64(0.37920088514346545 + Float64(x * -0.025050834237766436))))))
end

MATLAB

function tmp = code(x)
	tmp = x + (-1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436)))));
end

Wolfram

code[x_] := N[(x + N[(-1.0 / N[(0.4333638132548656 + N[(x * N[(0.37920088514346545 + N[(x * -0.025050834237766436), $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. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. +-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. fma-define 100.0%

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

   6. +-commutative 100.0%

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

   7. fma-define 100.0%

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

   8. +-commutative 100.0%

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

   9. fma-define 100.0%

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

4. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 99.0%

   \[\leadsto x - \frac{1}{\color{blue}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + -0.025050834237766436 \cdot x\right)}} \]
8. Step-by-step derivation

   1. *-commutative 99.0%

      \[\leadsto x - \frac{1}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + \color{blue}{x \cdot -0.025050834237766436}\right)} \]

9. Simplified 99.0%

   \[\leadsto x - \frac{1}{\color{blue}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + x \cdot -0.025050834237766436\right)}} \]

10. Final simplification 99.0%

    \[\leadsto x + \frac{-1}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + x \cdot -0.025050834237766436\right)} \]
11. Add Preprocessing

Alternative 4: 98.3% accurate, 1.5× 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. Add Preprocessing

   3. Taylor expanded in x around 0 98.3%

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

   4. Taylor expanded in x around inf 98.9%

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

   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. Add Preprocessing

   3. Taylor expanded in x around 0 98.6%

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

   4. Taylor expanded in x around 0 98.5%

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

Alternative 5: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ x + \frac{-1}{0.4333638132548656 + x \cdot 0.37920088514346545} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ x (/ -1.0 (+ 0.4333638132548656 (* x 0.37920088514346545)))))

C

double code(double x) {
	return x + (-1.0 / (0.4333638132548656 + (x * 0.37920088514346545)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((-1.0d0) / (0.4333638132548656d0 + (x * 0.37920088514346545d0)))
end function

Java

public static double code(double x) {
	return x + (-1.0 / (0.4333638132548656 + (x * 0.37920088514346545)));
}

Python

def code(x):
	return x + (-1.0 / (0.4333638132548656 + (x * 0.37920088514346545)))

Julia

function code(x)
	return Float64(x + Float64(-1.0 / Float64(0.4333638132548656 + Float64(x * 0.37920088514346545))))
end

MATLAB

function tmp = code(x)
	tmp = x + (-1.0 / (0.4333638132548656 + (x * 0.37920088514346545)));
end

Wolfram

code[x_] := N[(x + N[(-1.0 / N[(0.4333638132548656 + N[(x * 0.37920088514346545), $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. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. +-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. fma-define 100.0%

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

   6. +-commutative 100.0%

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

   7. fma-define 100.0%

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

   8. +-commutative 100.0%

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

   9. fma-define 100.0%

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

4. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 99.0%

   \[\leadsto x - \frac{1}{\color{blue}{0.4333638132548656 + 0.37920088514346545 \cdot x}} \]

8. Final simplification 99.0%

   \[\leadsto x + \frac{-1}{0.4333638132548656 + x \cdot 0.37920088514346545} \]
9. Add Preprocessing

Alternative 6: 97.8% 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 98.4%

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

Alternative 7: 50.8% 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 98.4%

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

4. Taylor expanded in x around 0 54.9%

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

Reproduce

herbie shell --seed 2024086 
(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)))))