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

Average Error: 0.0 → 0.0

Time: 6.7s

Precision: binary64

Cost: 7936

Math

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

↓

\[\frac{2.30753 + x \cdot 0.27061}{1 + \frac{x \cdot \left(0.9846394441 - {\left(x \cdot 0.04481\right)}^{2}\right)}{0.99229 + x \cdot -0.04481}} - x \]

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 (- 0.9846394441 (pow (* x 0.04481) 2.0)))
     (+ 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;
}

↓

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + ((x * (0.9846394441 - pow((x * 0.04481), 2.0))) / (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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + ((x * (0.9846394441d0 - ((x * 0.04481d0) ** 2.0d0))) / (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;
}

↓

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + ((x * (0.9846394441 - Math.pow((x * 0.04481), 2.0))) / (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

↓

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + ((x * (0.9846394441 - math.pow((x * 0.04481), 2.0))) / (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

↓

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(Float64(x * Float64(0.9846394441 - (Float64(x * 0.04481) ^ 2.0))) / 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

↓

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + ((x * (0.9846394441 - ((x * 0.04481) ^ 2.0))) / (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]

↓

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(x * N[(0.9846394441 - N[Power[N[(x * 0.04481), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.99229 + N[(x * -0.04481), $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

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\frac{\left(0.9846394441 - {\left(x \cdot 0.04481\right)}^{2}\right) \cdot x}{0.99229 + -0.04481 \cdot x}}} - x \]

3. Final simplification 0.0

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \frac{x \cdot \left(0.9846394441 - {\left(x \cdot 0.04481\right)}^{2}\right)}{0.99229 + x \cdot -0.04481}} - x \]

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	968

\[\begin{array}{l} t_0 := \left(\frac{6.039053782637804}{x} + \frac{\frac{-82.23527511657367}{x}}{x}\right) - x\\ \mathbf{if}\;x \leq -2644.5854058498144:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.3917278794254961:\\ \;\;\;\;2.30753 + x \cdot -3.0191289437\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.9
Cost	832

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

Alternative 4
Error	0.7
Cost	584

\[\begin{array}{l} t_0 := \frac{6.039053782637804}{x} - x\\ \mathbf{if}\;x \leq -2644.5854058498144:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.3917278794254961:\\ \;\;\;\;2.30753 + x \cdot -3.0191289437\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.1
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -2644.5854058498144:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.3917278794254961:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 6
Error	1.4
Cost	192

\[2.30753 - x \]

Alternative 7
Error	57.7
Cost	64

\[0.2727126142559131 \]

Alternative 8
Error	31.1
Cost	64

\[2.30753 \]

Reproduce

herbie shell --seed 2022308 
(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))