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

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 4

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, 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 2: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.6) (- 0.0 x) (if (<= x 1.15) 2.30753 (- 0.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = 0.0 - x;
	} else if (x <= 1.15) {
		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 <= 1.15d0) 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 <= 1.15) {
		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 <= 1.15:
		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 <= 1.15)
		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 <= 1.15)
		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, 1.15], 2.30753, N[(0.0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000009 or 1.1499999999999999 < 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. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

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

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. --lowering--.f64 N/A

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

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

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

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

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

      12. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot x} \]
   6. 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 98.8%

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

   7. Simplified 98.8%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 98.8%

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

   9. Applied egg-rr 98.8%

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

   if -3.60000000000000009 < x < 1.1499999999999999

   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. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

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

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. --lowering--.f64 N/A

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

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

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

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

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

      12. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% 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. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

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

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

   7. remove-double-neg N/A

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

   8. sub-neg N/A

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

   9. --lowering--.f64 N/A

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

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

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

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

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

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 98.3%

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

Alternative 4: 51.2% 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. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

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

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

   7. remove-double-neg N/A

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

   8. sub-neg N/A

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

   9. --lowering--.f64 N/A

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

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

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

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

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

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 47.5%

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

Reproduce

herbie shell --seed 2024191 
(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))