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

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 5

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-*.f64 N/A

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

   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. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift--.f64 100.0

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 100.0

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 100.0

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lift-*.f64 N/A

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

   21. lower-fma.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{2.30753 + x \cdot 0.27061}{-1 - x \cdot \left(0.99229 + x \cdot 0.04481\right)}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;x - 0.2727126142559131\\ \mathbf{elif}\;t\_0 \leq -1:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x - 0.2727126142559131\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+
          x
          (/
           (+ 2.30753 (* x 0.27061))
           (- -1.0 (* x (+ 0.99229 (* x 0.04481))))))))
   (if (<= t_0 -10.0)
     (- x 0.2727126142559131)
     (if (<= t_0 -1.0) -2.30753 (- x 0.2727126142559131)))))

C

double code(double x) {
	double t_0 = x + ((2.30753 + (x * 0.27061)) / (-1.0 - (x * (0.99229 + (x * 0.04481)))));
	double tmp;
	if (t_0 <= -10.0) {
		tmp = x - 0.2727126142559131;
	} else if (t_0 <= -1.0) {
		tmp = -2.30753;
	} else {
		tmp = x - 0.2727126142559131;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((2.30753d0 + (x * 0.27061d0)) / ((-1.0d0) - (x * (0.99229d0 + (x * 0.04481d0)))))
    if (t_0 <= (-10.0d0)) then
        tmp = x - 0.2727126142559131d0
    else if (t_0 <= (-1.0d0)) then
        tmp = -2.30753d0
    else
        tmp = x - 0.2727126142559131d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x + ((2.30753 + (x * 0.27061)) / (-1.0 - (x * (0.99229 + (x * 0.04481)))));
	double tmp;
	if (t_0 <= -10.0) {
		tmp = x - 0.2727126142559131;
	} else if (t_0 <= -1.0) {
		tmp = -2.30753;
	} else {
		tmp = x - 0.2727126142559131;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x + ((2.30753 + (x * 0.27061)) / (-1.0 - (x * (0.99229 + (x * 0.04481)))))
	tmp = 0
	if t_0 <= -10.0:
		tmp = x - 0.2727126142559131
	elif t_0 <= -1.0:
		tmp = -2.30753
	else:
		tmp = x - 0.2727126142559131
	return tmp

Julia

function code(x)
	t_0 = Float64(x + Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(-1.0 - Float64(x * Float64(0.99229 + Float64(x * 0.04481))))))
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = Float64(x - 0.2727126142559131);
	elseif (t_0 <= -1.0)
		tmp = -2.30753;
	else
		tmp = Float64(x - 0.2727126142559131);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x + ((2.30753 + (x * 0.27061)) / (-1.0 - (x * (0.99229 + (x * 0.04481)))));
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = x - 0.2727126142559131;
	elseif (t_0 <= -1.0)
		tmp = -2.30753;
	else
		tmp = x - 0.2727126142559131;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -10.0], N[(x - 0.2727126142559131), $MachinePrecision], If[LessEqual[t$95$0, -1.0], -2.30753, N[(x - 0.2727126142559131), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64))) x)))) < -10 or -1 < (-.f64 x (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64))) x))))

   1. Initial program 100.0%

      \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 100.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 100.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 100.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   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. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*l* N/A

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

      8. lft-mult-inverse N/A

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

      9. metadata-eval 98.1

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

   7. Simplified 98.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto x - \color{blue}{\frac{27061}{99229}} \]
   9. Step-by-step derivation

   10. Simplified 98.1%

       \[\leadsto x - \color{blue}{0.2727126142559131} \]

   if -10 < (-.f64 x (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64))) x)))) < -1

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 97.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{2.30753 + x \cdot 0.27061}{-1 - x \cdot \left(0.99229 + x \cdot 0.04481\right)} \leq -10:\\ \;\;\;\;x - 0.2727126142559131\\ \mathbf{elif}\;x + \frac{2.30753 + x \cdot 0.27061}{-1 - x \cdot \left(0.99229 + x \cdot 0.04481\right)} \leq -1:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x - 0.2727126142559131\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   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. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift--.f64 100.0

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 100.0

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 100.0

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lift-*.f64 N/A

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

   21. lower-fma.f64 100.0

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

4. Applied egg-rr 100.0%

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

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. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. lft-mult-inverse N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-*l* N/A

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

   8. lft-mult-inverse N/A

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

   9. metadata-eval 98.5

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

7. Simplified 98.5%

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

Alternative 4: 97.7% 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. Taylor expanded in x around 0

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

5. Simplified 97.5%

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

Alternative 5: 50.4% 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. Simplified 50.3%

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

Reproduce

herbie shell --seed 2024207 
(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)))))