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

Average Accuracy: 100.0% → 100.0%

Time: 9.5s

Precision: binary64

Cost: 13760

Math

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

↓

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

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)) (- 2.30753 (* x -0.27061)))))

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)) * (2.30753 - (x * -0.27061)));
}

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

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

   [Start] 100.0	\[ x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
   frac-2neg [=>] 100.0	\[ x - \color{blue}{\frac{-\left(2.30753 + x \cdot 0.27061\right)}{-\left(1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x\right)}} \]
   clear-num [=>] 100.0	\[ x - \color{blue}{\frac{1}{\frac{-\left(1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x\right)}{-\left(2.30753 + x \cdot 0.27061\right)}}} \]
   associate-/r/ [=>] 100.0	\[ x - \color{blue}{\frac{1}{-\left(1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x\right)} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right)} \]
   *-un-lft-identity [=>] 100.0	\[ x - \frac{1}{-\color{blue}{1 \cdot \left(1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x\right)}} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) \]
   distribute-lft-neg-in [=>] 100.0	\[ x - \frac{1}{\color{blue}{\left(-1\right) \cdot \left(1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x\right)}} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) \]
   associate-/r* [=>] 100.0	\[ x - \color{blue}{\frac{\frac{1}{-1}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x}} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) \]
   metadata-eval [=>] 100.0	\[ x - \frac{\frac{1}{\color{blue}{-1}}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) \]
   metadata-eval [=>] 100.0	\[ x - \frac{\color{blue}{-1}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) \]
   +-commutative [=>] 100.0	\[ x - \frac{-1}{\color{blue}{\left(0.99229 + x \cdot 0.04481\right) \cdot x + 1}} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) \]
   *-commutative [=>] 100.0	\[ x - \frac{-1}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right)} + 1} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) \]
   fma-def [=>] 100.0	\[ x - \frac{-1}{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) \]
   +-commutative [=>] 100.0	\[ x - \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.04481 + 0.99229}, 1\right)} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) \]
   fma-def [=>] 100.0	\[ x - \frac{-1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)}, 1\right)} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) \]
   neg-sub0 [=>] 100.0	\[ x - \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \color{blue}{\left(0 - \left(2.30753 + x \cdot 0.27061\right)\right)} \]
   metadata-eval [<=] 100.0	\[ x - \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \left(\color{blue}{\log 1} - \left(2.30753 + x \cdot 0.27061\right)\right) \]
   associate--r+ [=>] 100.0	\[ x - \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \color{blue}{\left(\left(\log 1 - 2.30753\right) - x \cdot 0.27061\right)} \]
   metadata-eval [=>] 100.0	\[ x - \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \left(\left(\color{blue}{0} - 2.30753\right) - x \cdot 0.27061\right) \]
   metadata-eval [=>] 100.0	\[ x - \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \left(\color{blue}{-2.30753} - x \cdot 0.27061\right) \]

3. Final simplification 100.0%

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

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	98.5%
Cost	832

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

Alternative 3
Accuracy	98.3%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	97.8%
Cost	192

\[x + -2.30753 \]

Alternative 5
Accuracy	49.9%
Cost	64

\[-2.30753 \]

Reproduce

herbie shell --seed 2023140 
(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)))))