Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D

Average Accuracy: 100.0% → 100.0%

Time: 9.3s

Precision: binary64

Cost: 20032

Math

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

↓

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

FPCore

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

↓

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

C

double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
}

↓

double code(double x) {
	return x - (1.0 / (fma(x, fma(x, 0.04481, 0.99229), 1.0) / fma(x, 0.27061, 2.30753)));
}

Julia

function code(x)
	return Float64(x - Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(Float64(0.99229 + Float64(x * 0.04481)) * x))))
end

↓

function code(x)
	return Float64(x - Float64(1.0 / Float64(fma(x, fma(x, 0.04481, 0.99229), 1.0) / fma(x, 0.27061, 2.30753))))
end

Wolfram

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

↓

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

Derivation

1. Initial program 100.0%

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

2. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
   clear-num [=>] 100.0	\[ x - \color{blue}{\frac{1}{\frac{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x}{2.30753 + x \cdot 0.27061}}} \]
   inv-pow [=>] 100.0	\[ x - \color{blue}{{\left(\frac{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x}{2.30753 + x \cdot 0.27061}\right)}^{-1}} \]
   +-commutative [=>] 100.0	\[ x - {\left(\frac{\color{blue}{\left(0.99229 + x \cdot 0.04481\right) \cdot x + 1}}{2.30753 + x \cdot 0.27061}\right)}^{-1} \]
   *-commutative [=>] 100.0	\[ x - {\left(\frac{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right)} + 1}{2.30753 + x \cdot 0.27061}\right)}^{-1} \]
   fma-def [=>] 100.0	\[ x - {\left(\frac{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}}{2.30753 + x \cdot 0.27061}\right)}^{-1} \]
   +-commutative [=>] 100.0	\[ x - {\left(\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.04481 + 0.99229}, 1\right)}{2.30753 + x \cdot 0.27061}\right)}^{-1} \]
   fma-def [=>] 100.0	\[ x - {\left(\frac{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)}, 1\right)}{2.30753 + x \cdot 0.27061}\right)}^{-1} \]
   +-commutative [=>] 100.0	\[ x - {\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}{\color{blue}{x \cdot 0.27061 + 2.30753}}\right)}^{-1} \]
   fma-def [=>] 100.0	\[ x - {\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}\right)}^{-1} \]

3. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ x - {\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}\right)}^{-1} \]
   unpow-1 [=>] 100.0	\[ x - \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}} \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	1088

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

Alternative 2
Accuracy	99.1%
Cost	576

\[x - \frac{1}{0.4333638132548656 + x \cdot 0.37920088514346545} \]

Alternative 3
Accuracy	98.4%
Cost	328

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

Alternative 4
Accuracy	97.9%
Cost	192

\[x - 2.30753 \]

Alternative 5
Accuracy	50.5%
Cost	64

\[-2.30753 \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))