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

Average Error: 0.0 → 0.0

Time: 16.5s

Precision: binary64

Cost: 7488

Math

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

↓

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

FPCore

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

↓

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

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

3. Applied egg-rr 0.0

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

4. Applied egg-rr 0.0

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

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	0.9
Cost	832

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

Alternative 3
Error	1.1
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4
Error	1.4
Cost	192

\[2.30753 - x \]

Alternative 5
Error	32.3
Cost	64

\[2.30753 \]

Reproduce

herbie shell --seed 2023010 
(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))