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

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 6

Speedup: 0.7×

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)\\ x - \mathsf{fma}\left(x, \frac{0.27061}{t\_0}, \frac{2.30753}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fma 0.04481 x 0.99229) x 1.0)))
   (- x (fma x (/ 0.27061 t_0) (/ 2.30753 t_0)))))

C

double code(double x) {
	double t_0 = fma(fma(0.04481, x, 0.99229), x, 1.0);
	return x - fma(x, (0.27061 / t_0), (2.30753 / t_0));
}

Julia

function code(x)
	t_0 = fma(fma(0.04481, x, 0.99229), x, 1.0)
	return Float64(x - fma(x, Float64(0.27061 / t_0), Float64(2.30753 / t_0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x + 1.0), $MachinePrecision]}, N[(x - N[(x * N[(0.27061 / t$95$0), $MachinePrecision] + N[(2.30753 / t$95$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. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}} \]

   2. lift-+.f64 N/A

      \[\leadsto x - \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} \]

   3. +-commutative N/A

      \[\leadsto x - \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} \]

   4. div-add N/A

      \[\leadsto x - \color{blue}{\left(\frac{x \cdot \frac{27061}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + \frac{\frac{230753}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto x - \left(\frac{\color{blue}{x \cdot \frac{27061}{100000}}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + \frac{\frac{230753}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}\right) \]

   6. associate-/l* N/A

      \[\leadsto x - \left(\color{blue}{x \cdot \frac{\frac{27061}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}} + \frac{\frac{230753}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}\right) \]

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 N/A

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

   18. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x - N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x + 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto x - \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}} \]

   2. +-commutative N/A

      \[\leadsto x - \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x + 1}} \]

   3. lift-*.f64 N/A

      \[\leadsto x - \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} \]

   4. lower-fma.f64 100.0

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x - N[(N[(0.27061 * x + 2.30753), $MachinePrecision] / N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x + 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto x - \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} \]

   2. +-commutative N/A

      \[\leadsto x - \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} \]

   3. lift-*.f64 N/A

      \[\leadsto x - \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} \]

   4. *-commutative N/A

      \[\leadsto x - \frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} \]

   5. lower-fma.f64 100.0

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 100.0

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 4: 98.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x - N[(N[(x * 0.27061 + 2.30753), $MachinePrecision] / N[(0.99229 * x + 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. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto x - \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + \frac{99229}{100000} \cdot x}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x - \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\frac{99229}{100000} \cdot x + 1}} \]

   2. lower-fma.f64 99.2

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

5. Applied rewrites 99.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.2

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

7. Applied rewrites 99.2%

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

Alternative 5: 97.6% accurate, 9.8× 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. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}} \]

   2. lift-+.f64 N/A

      \[\leadsto x - \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} \]

   3. +-commutative N/A

      \[\leadsto x - \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} \]

   4. div-add N/A

      \[\leadsto x - \color{blue}{\left(\frac{x \cdot \frac{27061}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + \frac{\frac{230753}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto x - \left(\frac{\color{blue}{x \cdot \frac{27061}{100000}}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + \frac{\frac{230753}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}\right) \]

   6. associate-/l* N/A

      \[\leadsto x - \left(\color{blue}{x \cdot \frac{\frac{27061}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}} + \frac{\frac{230753}{100000}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x}\right) \]

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 N/A

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

   18. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.7%

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

Alternative 6: 51.4% accurate, 39.0× 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

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

5. Applied rewrites 54.5%

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

Reproduce

herbie shell --seed 2024320 
(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)))))