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

Average Error: 0.0 → 0.0

Time: 8.8s

Precision: binary64

Cost: 1216

Math

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

↓

\[x - \frac{2.30753 + x \cdot 0.27061}{1 + \frac{x}{\frac{1}{x \cdot 0.04481 + 0.99229}}} \]

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))
   (+ 1.0 (/ x (/ 1.0 (+ (* x 0.04481) 0.99229)))))))

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)) / (1.0 + (x / (1.0 / ((x * 0.04481) + 0.99229)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + ((0.99229d0 + (x * 0.04481d0)) * x)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x / (1.0d0 / ((x * 0.04481d0) + 0.99229d0)))))
end function

Java

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

↓

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

Python

def code(x):
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)))

↓

def code(x):
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + (x / (1.0 / ((x * 0.04481) + 0.99229)))))

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)) / Float64(1.0 + Float64(x / Float64(1.0 / Float64(Float64(x * 0.04481) + 0.99229))))))
end

MATLAB

function tmp = code(x)
	tmp = x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
end

↓

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

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

3. Applied egg-rr 0.0

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \frac{x}{\frac{1}{\color{blue}{x \cdot 0.04481 + 0.99229}}}} \]

4. Final simplification 0.0

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{1 + \frac{x}{\frac{1}{x \cdot 0.04481 + 0.99229}}} \]

Alternatives

Alternative 1
Error	0.0
Cost	1088

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

Alternative 2
Error	0.8
Cost	832

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

Alternative 3
Error	0.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	1.2
Cost	192

\[x + -2.30753 \]

Alternative 5
Error	31.9
Cost	64

\[-2.30753 \]

Reproduce

herbie shell --seed 2023039 
(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)))))