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

Average Accuracy: 100.0% → 100.0%

Time: 9.0s

Precision: binary64

Cost: 1472

Math

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

↓

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

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) (+ (+ 1.0 (* x (* x 0.04481))) -1.0))))
  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) + ((1.0 + (x * (x * 0.04481))) + -1.0)))) - 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) + ((1.0d0 + (x * (x * 0.04481d0))) + (-1.0d0))))) - 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) + ((1.0 + (x * (x * 0.04481))) + -1.0)))) - 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) + ((1.0 + (x * (x * 0.04481))) + -1.0)))) - 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(Float64(x * 0.99229) + Float64(Float64(1.0 + Float64(x * Float64(x * 0.04481))) + -1.0)))) - 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) + ((1.0 + (x * (x * 0.04481))) + -1.0)))) - 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[(N[(x * 0.99229), $MachinePrecision] + N[(N[(1.0 + N[(x * N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   expm1-log1p-u [=>] 99.9	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(0.99229 + x \cdot 0.04481\right)\right)\right)}} - x \]
   log1p-def [<=] 99.9	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)\right)}\right)} - x \]
   expm1-udef [=>] 99.9	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\left(e^{\log \left(1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)\right)} - 1\right)}} - x \]
   add-exp-log [<=] 100.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \left(\color{blue}{\left(1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)\right)} - 1\right)} - x \]
   +-commutative [=>] 100.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \left(\color{blue}{\left(x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1\right)} - 1\right)} - x \]
   distribute-rgt-in [=>] 100.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \left(\left(\color{blue}{\left(0.99229 \cdot x + \left(x \cdot 0.04481\right) \cdot x\right)} + 1\right) - 1\right)} - x \]
   associate-+l+ [=>] 100.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \left(\color{blue}{\left(0.99229 \cdot x + \left(\left(x \cdot 0.04481\right) \cdot x + 1\right)\right)} - 1\right)} - x \]
   associate--l+ [=>] 100.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\left(0.99229 \cdot x + \left(\left(\left(x \cdot 0.04481\right) \cdot x + 1\right) - 1\right)\right)}} - x \]
   *-commutative [=>] 100.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \left(\color{blue}{x \cdot 0.99229} + \left(\left(\left(x \cdot 0.04481\right) \cdot x + 1\right) - 1\right)\right)} - x \]
   +-commutative [=>] 100.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \left(x \cdot 0.99229 + \left(\color{blue}{\left(1 + \left(x \cdot 0.04481\right) \cdot x\right)} - 1\right)\right)} - x \]
   *-commutative [=>] 100.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + \left(x \cdot 0.99229 + \left(\left(1 + \color{blue}{x \cdot \left(x \cdot 0.04481\right)}\right) - 1\right)\right)} - x \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	1088

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

Alternative 2
Accuracy	98.6%
Cost	832

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

Alternative 3
Accuracy	98.4%
Cost	392

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

Alternative 4
Accuracy	97.9%
Cost	192

\[2.30753 - x \]

Alternative 5
Accuracy	51.2%
Cost	64

\[2.30753 \]

Reproduce

herbie shell --seed 2023133 
(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))