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

Percentage Accurate: 100.0% → 100.0%

Time: 7.7s

Alternatives: 8

Speedup: 1.2×

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x - N[(N[(x * 0.27061 + 2.30753), $MachinePrecision] / N[(N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x), $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. 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. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 100.0

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. lift-fma.f64 100.0

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

6. Applied rewrites 100.0%

   \[\leadsto x - \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x + 1}} \]
7. 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: 98.7% 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. 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. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. lft-mult-inverse N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-*l* N/A

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

   7. lft-mult-inverse N/A

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

   8. metadata-eval 98.4

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

7. Applied rewrites 98.4%

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

Alternative 5: 98.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x - N[(2.30753 / 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. Taylor expanded in x around 0

   \[\leadsto x - \frac{\color{blue}{\frac{230753}{100000}}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} \]
4. Step-by-step derivation

5. Applied rewrites 98.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

7. Applied rewrites 98.0%

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

Alternative 6: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x - N[(2.30753 / 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

   \[\leadsto x - \frac{\color{blue}{\frac{230753}{100000}}}{1 + \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} \]
4. Step-by-step derivation

5. Applied rewrites 98.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. lft-mult-inverse N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. associate-*l* N/A

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

   10. lft-mult-inverse N/A

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

   11. metadata-eval 98.0

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

8. Applied rewrites 98.0%

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

9. Final simplification 98.0%

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

Alternative 7: 98.0% 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

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

   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. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 97.7%

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

Alternative 8: 50.1% 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 51.8%

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

Reproduce

herbie shell --seed 2024313 
(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)))))