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

Average Accuracy: 100.0% → 100.0%

Time: 9.2s

Precision: binary64

Cost: 1344

Math

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

↓

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

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

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 + (1.0d0 + ((x * ((x * 0.04481d0) + 0.99229d0)) + (-1.0d0)))))
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 + (1.0 + ((x * ((x * 0.04481) + 0.99229)) + -1.0))));
}

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

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

   [Start] 100.0	\[ x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
   expm1-log1p-u [=>] 99.9	\[ x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.99229 + x \cdot 0.04481\right) \cdot x\right)\right)}} \]
   log1p-def [<=] 99.9	\[ x - \frac{2.30753 + x \cdot 0.27061}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x\right)}\right)} \]
   expm1-udef [=>] 99.9	\[ x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\left(e^{\log \left(1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x\right)} - 1\right)}} \]
   add-exp-log [<=] 100.0	\[ x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(\color{blue}{\left(1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x\right)} - 1\right)} \]
   associate--l+ [=>] 100.0	\[ x - \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\left(1 + \left(\left(0.99229 + x \cdot 0.04481\right) \cdot x - 1\right)\right)}} \]
   *-commutative [=>] 100.0	\[ x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(1 + \left(\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right)} - 1\right)\right)} \]
   +-commutative [=>] 100.0	\[ x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(1 + \left(x \cdot \color{blue}{\left(x \cdot 0.04481 + 0.99229\right)} - 1\right)\right)} \]
   fma-def [=>] 100.0	\[ x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(1 + \left(x \cdot \color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)} - 1\right)\right)} \]

3. Applied egg-rr 100.0%

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

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

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	1088

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

Alternative 2
Accuracy	98.6%
Cost	832

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

Alternative 3
Accuracy	98.4%
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
Accuracy	97.9%
Cost	192

\[x + -2.30753 \]

Alternative 5
Accuracy	51.2%
Cost	64

\[-2.30753 \]

Reproduce

herbie shell --seed 2023133 
(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)))))