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

Average Error: 0.02% → 0.02%

Time: 10.0s

Precision: binary64

Cost: 7488

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (-
  (*
   (/ -1.0 (fma 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 / fma(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(Float64(-1.0 / fma(x, Float64(Float64(x * 0.04481) + 0.99229), 1.0)) * Float64(-2.30753 + Float64(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[(N[(-1.0 / N[(x * N[(N[(x * 0.04481), $MachinePrecision] + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(-2.30753 + N[(x * -0.27061), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 0.02

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

2. Applied egg-rr 0.02

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

3. Applied egg-rr 0.02

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

4. Final simplification 0.02

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

Alternatives

Alternative 1
Error	0.02%
Cost	1088

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

Alternative 2
Error	1.33%
Cost	832

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

Alternative 3
Error	1.65%
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -3.7:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4
Error	2.08%
Cost	192

\[2.30753 - x \]

Alternative 5
Error	90.27%
Cost	64

\[0.2727126142559131 \]

Alternative 6
Error	49.27%
Cost	64

\[2.30753 \]

Reproduce

herbie shell --seed 2023088 
(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))