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

Percentage Accurate: 100.0% → 100.0%

Time: 10.5s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. neg-mul-1 100.0%

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

   3. *-commutative 100.0%

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

   4. cancel-sign-sub 100.0%

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

   5. *-commutative 100.0%

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

   6. neg-mul-1 100.0%

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

   7. remove-double-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. fma-define 100.0%

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

   10. +-commutative 100.0%

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

   11. *-commutative 100.0%

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

   12. fma-define 100.0%

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

   13. +-commutative 100.0%

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

   14. fma-define 100.0%

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

3. Simplified 100.0%

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

   1. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. div-inv 100.0%

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

   4. fma-undefine 100.0%

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

   5. +-commutative 100.0%

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

   6. fma-undefine 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-undefine 100.0%

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

   9. +-commutative 100.0%

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

   10. +-commutative 100.0%

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

   11. distribute-rgt-neg-in 100.0%

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

   12. fma-define 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. neg-mul-1 100.0%

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

   3. *-commutative 100.0%

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

   4. cancel-sign-sub 100.0%

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

   5. *-commutative 100.0%

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

   6. neg-mul-1 100.0%

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

   7. remove-double-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. fma-define 100.0%

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

   10. +-commutative 100.0%

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

   11. *-commutative 100.0%

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

   12. fma-define 100.0%

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

   13. +-commutative 100.0%

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

   14. fma-define 100.0%

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

3. Simplified 100.0%

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

   1. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. div-inv 100.0%

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

   4. fma-undefine 100.0%

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

   5. +-commutative 100.0%

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

   6. fma-undefine 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-undefine 100.0%

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

   9. +-commutative 100.0%

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

   10. +-commutative 100.0%

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

   11. distribute-rgt-neg-in 100.0%

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

   12. fma-define 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 100.0%

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

   1. fma-undefine 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(2.30753 + x \cdot 0.27061, \frac{1}{-1 - x \cdot \left(0.99229 + x \cdot 0.04481\right)}, x\right) \]
11. 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: 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 98.6%

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

   1. *-commutative 98.6%

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

5. Simplified 98.6%

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

6. Final simplification 98.6%

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

Alternative 5: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05) x (if (<= x 1.15) -2.30753 x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		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 <= (-1.05d0)) 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 <= -1.05) {
		tmp = x;
	} else if (x <= 1.15) {
		tmp = -2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = x
	elif x <= 1.15:
		tmp = -2.30753
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		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 <= -1.05)
		tmp = x;
	elseif (x <= 1.15)
		tmp = -2.30753;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], x, If[LessEqual[x, 1.15], -2.30753, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 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. sub-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. *-commutative 100.0%

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

      4. cancel-sign-sub 100.0%

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

      5. *-commutative 100.0%

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

      6. neg-mul-1 100.0%

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

      7. remove-double-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. fma-define 100.0%

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

      10. +-commutative 100.0%

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

      11. *-commutative 100.0%

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

      12. fma-define 100.0%

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

      13. +-commutative 100.0%

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

      14. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.2%

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

   if -1.05000000000000004 < 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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. neg-mul-1 99.9%

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

      3. *-commutative 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. *-commutative 99.9%

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

      6. neg-mul-1 99.9%

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

      7. remove-double-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-define 99.9%

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

      10. +-commutative 99.9%

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

      11. *-commutative 99.9%

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

      12. fma-define 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.4%

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

Alternative 6: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x + N[(2.30753 / 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 98.6%

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

   1. *-commutative 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in x around 0 98.5%

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

7. Final simplification 98.5%

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

Alternative 7: 98.0% 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. sub-neg 100.0%

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

   2. neg-mul-1 100.0%

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

   3. *-commutative 100.0%

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

   4. cancel-sign-sub 100.0%

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

   5. *-commutative 100.0%

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

   6. neg-mul-1 100.0%

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

   7. remove-double-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. fma-define 100.0%

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

   10. +-commutative 100.0%

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

   11. *-commutative 100.0%

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

   12. fma-define 100.0%

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

   13. +-commutative 100.0%

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

   14. fma-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 58.5%

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

   1. *-commutative 58.5%

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

7. Simplified 58.5%

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

8. Taylor expanded in x around 0 97.7%

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

Alternative 8: 50.0% 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. sub-neg 100.0%

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

   2. neg-mul-1 100.0%

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

   3. *-commutative 100.0%

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

   4. cancel-sign-sub 100.0%

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

   5. *-commutative 100.0%

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

   6. neg-mul-1 100.0%

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

   7. remove-double-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. fma-define 100.0%

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

   10. +-commutative 100.0%

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

   11. *-commutative 100.0%

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

   12. fma-define 100.0%

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

   13. +-commutative 100.0%

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

   14. fma-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 50.7%

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

Reproduce

herbie shell --seed 2024144 
(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)))))