Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

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

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

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

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(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] - x), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

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

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

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

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(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] - x), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (((x * ((x * (-0.04481d0)) + (-0.99229d0))) + (-1.0d0)) / ((x * (-0.27061d0)) + (-2.30753d0)))) - x
end function

Java

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

Python

def code(x):
	return (1.0 / (((x * ((x * -0.04481) + -0.99229)) + -1.0) / ((x * -0.27061) + -2.30753))) - x

Julia

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

MATLAB

function tmp = code(x)
	tmp = (1.0 / (((x * ((x * -0.04481) + -0.99229)) + -1.0) / ((x * -0.27061) + -2.30753))) - x;
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

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

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

   3. frac-2neg N/A

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

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

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

   5. +-commutative N/A

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

   6. distribute-neg-in N/A

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

   7. metadata-eval N/A

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

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

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

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

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

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

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. +-commutative N/A

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

4. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

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

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

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

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(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] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

Alternative 3: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.6) (- 0.0 x) (if (<= x 1.2) 2.30753 (- 0.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = 0.0 - x;
	} else if (x <= 1.2) {
		tmp = 2.30753;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.6d0)) then
        tmp = 0.0d0 - x
    else if (x <= 1.2d0) then
        tmp = 2.30753d0
    else
        tmp = 0.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = 0.0 - x;
	} else if (x <= 1.2) {
		tmp = 2.30753;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.6:
		tmp = 0.0 - x
	elif x <= 1.2:
		tmp = 2.30753
	else:
		tmp = 0.0 - x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.6)
		tmp = Float64(0.0 - x);
	elseif (x <= 1.2)
		tmp = 2.30753;
	else
		tmp = Float64(0.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.6)
		tmp = 0.0 - x;
	elseif (x <= 1.2)
		tmp = 2.30753;
	else
		tmp = 0.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.6], N[(0.0 - x), $MachinePrecision], If[LessEqual[x, 1.2], 2.30753, N[(0.0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000009 or 1.19999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x} \]

      3. --lowering--.f64 98.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{x}\right) \]

   5. Simplified 98.6%

      \[\leadsto \color{blue}{0 - x} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(x\right) \]

      2. neg-lowering-neg.f64 98.6%

         \[\leadsto \mathsf{neg.f64}\left(x\right) \]

   7. Applied egg-rr 98.6%

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

   if -3.60000000000000009 < x < 1.19999999999999996

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 97.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + x \cdot -0.025050834237766436\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  (/
   1.0
   (+
    0.4333638132548656
    (* x (+ 0.37920088514346545 (* x -0.025050834237766436)))))
  x))

C

double code(double x) {
	return (1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (0.4333638132548656d0 + (x * (0.37920088514346545d0 + (x * (-0.025050834237766436d0)))))) - x
end function

Java

public static double code(double x) {
	return (1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436))))) - x;
}

Python

def code(x):
	return (1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436))))) - x

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(0.4333638132548656 + Float64(x * Float64(0.37920088514346545 + Float64(x * -0.025050834237766436))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436))))) - x;
end

Wolfram

code[x_] := N[(N[(1.0 / N[(0.4333638132548656 + N[(x * N[(0.37920088514346545 + N[(x * -0.025050834237766436), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

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

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

   3. frac-2neg N/A

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

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

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

   5. +-commutative N/A

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

   6. distribute-neg-in N/A

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

   7. metadata-eval N/A

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

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

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

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

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

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

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. +-commutative N/A

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(\frac{100000}{230753} + x \cdot \left(\frac{20191289437}{53246947009} + \frac{-307796913907328}{12286892763167777} \cdot x\right)\right)}\right), x\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{100000}{230753}, \left(x \cdot \left(\frac{20191289437}{53246947009} + \frac{-307796913907328}{12286892763167777} \cdot x\right)\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{100000}{230753}, \mathsf{*.f64}\left(x, \left(\frac{20191289437}{53246947009} + \frac{-307796913907328}{12286892763167777} \cdot x\right)\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{100000}{230753}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{20191289437}{53246947009}, \left(\frac{-307796913907328}{12286892763167777} \cdot x\right)\right)\right)\right)\right), x\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{100000}{230753}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{20191289437}{53246947009}, \left(x \cdot \frac{-307796913907328}{12286892763167777}\right)\right)\right)\right)\right), x\right) \]

   5. *-lowering-*.f64 98.7%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{100000}{230753}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{20191289437}{53246947009}, \mathsf{*.f64}\left(x, \frac{-307796913907328}{12286892763167777}\right)\right)\right)\right)\right), x\right) \]

7. Simplified 98.7%

   \[\leadsto \frac{1}{\color{blue}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + x \cdot -0.025050834237766436\right)}} - x \]
8. Add Preprocessing

Alternative 5: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{0.4333638132548656 + x \cdot 0.37920088514346545} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ 1.0 (+ 0.4333638132548656 (* x 0.37920088514346545))) x))

C

double code(double x) {
	return (1.0 / (0.4333638132548656 + (x * 0.37920088514346545))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (0.4333638132548656d0 + (x * 0.37920088514346545d0))) - x
end function

Java

public static double code(double x) {
	return (1.0 / (0.4333638132548656 + (x * 0.37920088514346545))) - x;
}

Python

def code(x):
	return (1.0 / (0.4333638132548656 + (x * 0.37920088514346545))) - x

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(0.4333638132548656 + Float64(x * 0.37920088514346545))) - x)
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (0.4333638132548656 + (x * 0.37920088514346545))) - x;
end

Wolfram

code[x_] := N[(N[(1.0 / N[(0.4333638132548656 + N[(x * 0.37920088514346545), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

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

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

   3. frac-2neg N/A

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

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

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

   5. +-commutative N/A

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

   6. distribute-neg-in N/A

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

   7. metadata-eval N/A

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

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

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

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

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

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

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. +-commutative N/A

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(\frac{100000}{230753} + \frac{20191289437}{53246947009} \cdot x\right)}\right), x\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{100000}{230753}, \left(\frac{20191289437}{53246947009} \cdot x\right)\right)\right), x\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{100000}{230753}, \left(x \cdot \frac{20191289437}{53246947009}\right)\right)\right), x\right) \]

   3. *-lowering-*.f64 98.6%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{100000}{230753}, \mathsf{*.f64}\left(x, \frac{20191289437}{53246947009}\right)\right)\right), x\right) \]

7. Simplified 98.6%

   \[\leadsto \frac{1}{\color{blue}{0.4333638132548656 + x \cdot 0.37920088514346545}} - x \]
8. Add Preprocessing

Alternative 6: 98.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ 2.30753 - x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 2.30753 x))

C

double code(double x) {
	return 2.30753 - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.30753d0 - x
end function

Java

public static double code(double x) {
	return 2.30753 - x;
}

Python

def code(x):
	return 2.30753 - x

Julia

function code(x)
	return Float64(2.30753 - x)
end

MATLAB

function tmp = code(x)
	tmp = 2.30753 - x;
end

Wolfram

code[x_] := N[(2.30753 - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 97.6%

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

Alternative 7: 50.9% 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%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.0%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))