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

Average Error: 0.02% → 0.02%

Time: 9.0s

Precision: binary64

Cost: 7360

Math

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

↓

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

FPCore

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

↓

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

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 + ((-2.30753 + (x * -0.27061)) / fma(x, ((x * 0.04481) + 0.99229), 1.0));
}

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

Derivation

1. Initial program 0.02

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

2. Simplified 0.02

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

   [Start] 0.02	\[ x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
   +-commutative [=>] 0.02	\[ x - \frac{\color{blue}{x \cdot 0.27061 + 2.30753}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
   fma-def [=>] 0.02	\[ x - \frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
   +-commutative [=>] 0.02	\[ x - \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{\left(0.99229 + x \cdot 0.04481\right) \cdot x + 1}} \]
   *-commutative [=>] 0.02	\[ x - \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right)} + 1} \]
   fma-def [=>] 0.02	\[ x - \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}} \]
   +-commutative [=>] 0.02	\[ x - \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.04481 + 0.99229}, 1\right)} \]
   fma-def [=>] 0.02	\[ x - \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)}, 1\right)} \]

3. Applied egg-rr 0.02

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

4. Applied egg-rr 0.02

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

5. Final simplification 0.02

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

Alternatives

Alternative 1
Error	0.02%
Cost	1088

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

Alternative 2
Error	1.16%
Cost	832

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

Alternative 3
Error	1.49%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	1.9%
Cost	192

\[x + -2.30753 \]

Alternative 5
Error	48.87%
Cost	64

\[-2.30753 \]

Reproduce

herbie shell --seed 2023121 
(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)))))