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

Average Error: 0.0 → 0.0

Time: 7.6s

Precision: binary64

Cost: 1600

Math

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

↓

\[\frac{2.30753 + x \cdot 0.27061}{1 + \frac{x}{\frac{0.99229 + x \cdot -0.04481}{0.9846394441 + -0.0020079361 \cdot \left(x \cdot x\right)}}} - x \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (-
  (/
   (+ 2.30753 (* x 0.27061))
   (+
    1.0
    (/
     x
     (/
      (+ 0.99229 (* x -0.04481))
      (+ 0.9846394441 (* -0.0020079361 (* x x)))))))
  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 ((2.30753 + (x * 0.27061)) / (1.0 + (x / ((0.99229 + (x * -0.04481)) / (0.9846394441 + (-0.0020079361 * (x * x))))))) - 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 = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x / ((0.99229d0 + (x * (-0.04481d0))) / (0.9846394441d0 + ((-0.0020079361d0) * (x * x))))))) - 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 ((2.30753 + (x * 0.27061)) / (1.0 + (x / ((0.99229 + (x * -0.04481)) / (0.9846394441 + (-0.0020079361 * (x * x))))))) - x;
}

Python

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

↓

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x / ((0.99229 + (x * -0.04481)) / (0.9846394441 + (-0.0020079361 * (x * x))))))) - 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(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x / Float64(Float64(0.99229 + Float64(x * -0.04481)) / Float64(0.9846394441 + Float64(-0.0020079361 * Float64(x * x))))))) - 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 = ((2.30753 + (x * 0.27061)) / (1.0 + (x / ((0.99229 + (x * -0.04481)) / (0.9846394441 + (-0.0020079361 * (x * x))))))) - 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[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x / N[(N[(0.99229 + N[(x * -0.04481), $MachinePrecision]), $MachinePrecision] / N[(0.9846394441 + N[(-0.0020079361 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\frac{0.9846394441 - 0.0020079361 \cdot \left(x \cdot x\right)}{\frac{0.99229 + -0.04481 \cdot x}{x}}}} - x \]

3. Simplified 0.0

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\frac{x}{\frac{0.99229 + x \cdot -0.04481}{0.9846394441 + -0.0020079361 \cdot \left(x \cdot x\right)}}}} - x \]
   Proof

   [Start] 0.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \frac{0.9846394441 - 0.0020079361 \cdot \left(x \cdot x\right)}{\frac{0.99229 + -0.04481 \cdot x}{x}}} - x \]
   associate-/l* [<=] 0.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\frac{\left(0.9846394441 - 0.0020079361 \cdot \left(x \cdot x\right)\right) \cdot x}{0.99229 + -0.04481 \cdot x}}} - x \]
   *-commutative [<=] 0.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\color{blue}{x \cdot \left(0.9846394441 - 0.0020079361 \cdot \left(x \cdot x\right)\right)}}{0.99229 + -0.04481 \cdot x}} - x \]
   *-rgt-identity [<=] 0.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \frac{x \cdot \left(0.9846394441 - 0.0020079361 \cdot \left(x \cdot x\right)\right)}{\color{blue}{\left(0.99229 + -0.04481 \cdot x\right) \cdot 1}}} - x \]
   associate-/l* [=>] 0.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\frac{x}{\frac{\left(0.99229 + -0.04481 \cdot x\right) \cdot 1}{0.9846394441 - 0.0020079361 \cdot \left(x \cdot x\right)}}}} - x \]
   *-rgt-identity [=>] 0.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \frac{x}{\frac{\color{blue}{0.99229 + -0.04481 \cdot x}}{0.9846394441 - 0.0020079361 \cdot \left(x \cdot x\right)}}} - x \]
   *-commutative [=>] 0.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \frac{x}{\frac{0.99229 + \color{blue}{x \cdot -0.04481}}{0.9846394441 - 0.0020079361 \cdot \left(x \cdot x\right)}}} - x \]
   cancel-sign-sub-inv [=>] 0.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \frac{x}{\frac{0.99229 + x \cdot -0.04481}{\color{blue}{0.9846394441 + \left(-0.0020079361\right) \cdot \left(x \cdot x\right)}}}} - x \]
   metadata-eval [=>] 0.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \frac{x}{\frac{0.99229 + x \cdot -0.04481}{0.9846394441 + \color{blue}{-0.0020079361} \cdot \left(x \cdot x\right)}}} - x \]

4. Final simplification 0.0

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \frac{x}{\frac{0.99229 + x \cdot -0.04481}{0.9846394441 + -0.0020079361 \cdot \left(x \cdot x\right)}}} - x \]

Alternatives

Alternative 1
Error	0.0
Cost	1088

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

Alternative 2
Error	0.8
Cost	832

\[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot 0.99229} - x \]

Alternative 3
Error	0.8
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4
Error	1.2
Cost	192

\[2.30753 - x \]

Alternative 5
Error	31.9
Cost	64

\[2.30753 \]

Reproduce

herbie shell --seed 2023039 
(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))