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

Average Error: 0.0 → 0.0

Time: 1.3min

Precision: binary64

Cost: 1344

Math

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

↓

\[\frac{2.30753}{\left(1 + \left(x \cdot 0.04481\right) \cdot x\right) + 0.99229 \cdot x} \cdot \left(\left(8.527142382025794 + x\right) \cdot 0.11727258150489918\right) - x \]

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 (+ (+ 1.0 (* (* x 0.04481) x)) (* 0.99229 x)))
   (* (+ 8.527142382025794 x) 0.11727258150489918))
  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 / ((1.0 + ((x * 0.04481) * x)) + (0.99229 * x))) * ((8.527142382025794 + x) * 0.11727258150489918)) - 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 / ((1.0d0 + ((x * 0.04481d0) * x)) + (0.99229d0 * x))) * ((8.527142382025794d0 + x) * 0.11727258150489918d0)) - 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 / ((1.0 + ((x * 0.04481) * x)) + (0.99229 * x))) * ((8.527142382025794 + x) * 0.11727258150489918)) - 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 / ((1.0 + ((x * 0.04481) * x)) + (0.99229 * x))) * ((8.527142382025794 + x) * 0.11727258150489918)) - 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(Float64(1.0 + Float64(Float64(x * 0.04481) * x)) + Float64(0.99229 * x))) * Float64(Float64(8.527142382025794 + x) * 0.11727258150489918)) - 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 / ((1.0 + ((x * 0.04481) * x)) + (0.99229 * x))) * ((8.527142382025794 + x) * 0.11727258150489918)) - 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[(N[(1.0 + N[(N[(x * 0.04481), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(0.99229 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(8.527142382025794 + x), $MachinePrecision] * 0.11727258150489918), $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

   \[\leadsto \color{blue}{2.30753 \cdot \frac{\frac{8.527142382025794 + x}{8.527142382025794}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} - x \]

3. Simplified 0.0

   \[\leadsto \color{blue}{\frac{2.30753}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \left(\left(8.527142382025794 + x\right) \cdot 0.11727258150489918\right)} - x \]
   Proof

4. Applied egg-rr 0.0

   \[\leadsto \frac{2.30753}{\color{blue}{\left(1 + \left(x \cdot 0.04481\right) \cdot x\right) + 0.99229 \cdot x}} \cdot \left(\left(8.527142382025794 + x\right) \cdot 0.11727258150489918\right) - x \]

Alternatives

Alternative 1
Error	0.0
Cost	1216

\[\frac{\left(8.527142382025794 + x\right) \cdot 0.11727258150489918}{\frac{-1 - x \cdot \left(x \cdot 0.04481 + 0.99229\right)}{-2.30753}} - x \]

Alternative 2
Error	0.5
Cost	1088

\[\frac{0.27061 \cdot \left(8.527142382025794 + x\right)}{1 + x \cdot \left(x \cdot 0.04481 + 0.99229\right)} - x \]

Alternative 3
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 4
Error	1.1
Cost	832

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

Alternative 5
Error	1.2
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.16:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 6
Error	1.5
Cost	192

\[2.30753 - x \]

Alternative 7
Error	31.9
Cost	64

\[2.30753 \]

Reproduce

herbie shell --seed 2023033 
(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))