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

Average Error: 0.0 → 0.0

Time: 1.7min

Precision: binary64

Cost: 20352

Math

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

↓

\[x - \frac{\mathsf{fma}\left(x, 0.11727258150489918, 1\right)}{\mathsf{fma}\left(0.99229, x, 1\right) \cdot 0.4333638132548656 + \left(0.04481 \cdot {x}^{2}\right) \cdot 0.4333638132548656} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (-
  x
  (/
   (fma x 0.11727258150489918 1.0)
   (+
    (* (fma 0.99229 x 1.0) 0.4333638132548656)
    (* (* 0.04481 (pow x 2.0)) 0.4333638132548656)))))

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 - (fma(x, 0.11727258150489918, 1.0) / ((fma(0.99229, x, 1.0) * 0.4333638132548656) + ((0.04481 * pow(x, 2.0)) * 0.4333638132548656)));
}

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(fma(x, 0.11727258150489918, 1.0) / Float64(Float64(fma(0.99229, x, 1.0) * 0.4333638132548656) + Float64(Float64(0.04481 * (x ^ 2.0)) * 0.4333638132548656))))
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[(x * 0.11727258150489918 + 1.0), $MachinePrecision] / N[(N[(N[(0.99229 * x + 1.0), $MachinePrecision] * 0.4333638132548656), $MachinePrecision] + N[(N[(0.04481 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] * 0.4333638132548656), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

   \[\leadsto x - \color{blue}{\frac{\mathsf{fma}\left(x, 0.11727258150489918, 1\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)}{2.30753}}} \]

3. Applied egg-rr 0.0

   \[\leadsto x - \frac{\mathsf{fma}\left(x, 0.11727258150489918, 1\right)}{\frac{\color{blue}{\left(1 + \left(0.04481 \cdot x\right) \cdot x\right) + 0.99229 \cdot x}}{2.30753}} \]

4. Applied egg-rr 0.0

   \[\leadsto x - \frac{\mathsf{fma}\left(x, 0.11727258150489918, 1\right)}{\color{blue}{\mathsf{fma}\left(0.99229, x, 1\right) \cdot 0.4333638132548656 + \left(0.04481 \cdot {x}^{2}\right) \cdot 0.4333638132548656}} \]

Alternatives

Alternative 1
Error	0.0
Cost	7616

\[x - \frac{\mathsf{fma}\left(x, 0.11727258150489918, 1\right)}{\frac{\left(1 + \left(0.04481 \cdot x\right) \cdot x\right) + 0.99229 \cdot x}{2.30753}} \]

Alternative 2
Error	0.0
Cost	1088

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

Alternative 3
Error	1.1
Cost	576

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

Alternative 4
Error	1.2
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.16:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	1.5
Cost	320

\[\left(-5.30753 + x\right) - -3 \]

Alternative 6
Error	1.5
Cost	192

\[x - 2.30753 \]

Alternative 7
Error	31.9
Cost	64

\[-2.30753 \]

Reproduce

herbie shell --seed 2023033 
(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)))))