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

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

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: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x + N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(N[(x * -0.99229), $MachinePrecision] + -1.0), $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. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right), \color{blue}{\left(1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \left(x \cdot \frac{27061}{100000}\right)\right), \left(\color{blue}{1} + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right) \]

   5. remove-double-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)}\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)\right)\right)\right) \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}\right)\right)\right)\right) \]

   11. distribute-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{99229}{100000}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{99229}{100000}\right)\right), \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{4481}{100000}}\right)\right)\right)\right)\right)\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   16. metadata-eval 100.0%

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \frac{-4481}{100000}\right)\right)\right)\right)\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{-99229}{100000} \cdot x\right)}\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\frac{-99229}{100000}}\right)\right)\right)\right) \]

   2. *-lowering-*.f64 98.9%

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{-99229}{100000}}\right)\right)\right)\right) \]

7. Simplified 98.9%

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

8. Final simplification 98.9%

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

Alternative 3: 98.4% 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. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right), \color{blue}{\left(1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \left(x \cdot \frac{27061}{100000}\right)\right), \left(\color{blue}{1} + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right) \]

      5. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)}\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)\right)\right)\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}\right)\right)\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{99229}{100000}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{99229}{100000}\right)\right), \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{4481}{100000}}\right)\right)\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      16. metadata-eval 100.0%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \frac{-4481}{100000}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   6. Step-by-step derivation

   7. Simplified 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. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right), \color{blue}{\left(1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \left(x \cdot \frac{27061}{100000}\right)\right), \left(\color{blue}{1} + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right) \]

      5. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)}\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)\right)\right)\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}\right)\right)\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{99229}{100000}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{99229}{100000}\right)\right), \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{4481}{100000}}\right)\right)\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      16. metadata-eval 100.0%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \frac{-4481}{100000}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-230753}{100000}} \]
   6. Step-by-step derivation

   7. Simplified 98.0%

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

Alternative 4: 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. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right), \color{blue}{\left(1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \left(x \cdot \frac{27061}{100000}\right)\right), \left(\color{blue}{1} + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right) \]

   5. remove-double-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)}\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)\right)\right)\right) \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}\right)\right)\right)\right) \]

   11. distribute-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{99229}{100000}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{99229}{100000}\right)\right), \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{4481}{100000}}\right)\right)\right)\right)\right)\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   16. metadata-eval 100.0%

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \frac{-4481}{100000}\right)\right)\right)\right)\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\frac{230753}{100000}}\right) \]
6. Step-by-step derivation

7. Simplified 98.3%

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

Alternative 5: 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. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right), \color{blue}{\left(1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \left(x \cdot \frac{27061}{100000}\right)\right), \left(\color{blue}{1} + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right) \]

   5. remove-double-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)}\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x\right)\right)}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)\right)\right)\right) \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)\right)\right)}\right)\right)\right)\right) \]

   11. distribute-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{99229}{100000}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{99229}{100000}\right)\right), \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{4481}{100000}}\right)\right)\right)\right)\right)\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{4481}{100000}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   16. metadata-eval 100.0%

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{230753}{100000}, \mathsf{*.f64}\left(x, \frac{27061}{100000}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-99229}{100000}, \mathsf{*.f64}\left(x, \frac{-4481}{100000}\right)\right)\right)\right)\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-230753}{100000}} \]
6. Step-by-step derivation

7. Simplified 51.7%

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

Reproduce

herbie shell --seed 2024161 
(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)))))