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

Average Error: 0.0 → 0.0

Time: 6.7s

Precision: binary64

Cost: 1344

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

↓

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

↓

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

Python

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

↓

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

↓

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

MATLAB

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

↓

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

↓

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

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Applied egg-rr 0.0

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

4. Applied egg-rr 0.0

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

5. Applied egg-rr 0.0

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

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	1088

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

Alternative 2
Error	0.6
Cost	576

\[\frac{1}{0.4333638132548656 + x \cdot 0.37920088514346545} - x \]

Alternative 3
Error	1.0
Cost	392

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

Alternative 4
Error	1.3
Cost	192

\[2.30753 - x \]

Alternative 5
Error	31.9
Cost	64

\[2.30753 \]

Reproduce

herbie shell --seed 2023016 
(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))