Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A

Percentage Accurate: 99.9% → 99.9%

Time: 6.4s

Precision: binary64

Cost: 704

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

↓

\[1 + \left(x \cdot \left(x \cdot -0.12\right) - 0.253 \cdot x\right) \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))

↓

(FPCore (x) :precision binary64 (+ 1.0 (- (* x (* x -0.12)) (* 0.253 x))))

C

double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

↓

double code(double x) {
	return 1.0 + ((x * (x * -0.12)) - (0.253 * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (0.253d0 + (x * 0.12d0)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((x * (x * (-0.12d0))) - (0.253d0 * x))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

↓

public static double code(double x) {
	return 1.0 + ((x * (x * -0.12)) - (0.253 * x));
}

Python

def code(x):
	return 1.0 - (x * (0.253 + (x * 0.12)))

↓

def code(x):
	return 1.0 + ((x * (x * -0.12)) - (0.253 * x))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12))))
end

↓

function code(x)
	return Float64(1.0 + Float64(Float64(x * Float64(x * -0.12)) - Float64(0.253 * x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (0.253 + (x * 0.12)));
end

↓

function tmp = code(x)
	tmp = 1.0 + ((x * (x * -0.12)) - (0.253 * x));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(1.0 + N[(N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision] - N[(0.253 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

2. Applied egg-rr 99.9%

   \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(x \cdot 0.12, x, x \cdot 0.253\right)} \]
   Step-by-step derivation

   [Start] 99.9%	\[ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   +-commutative [=>] 99.9%	\[ 1 - x \cdot \color{blue}{\left(x \cdot 0.12 + 0.253\right)} \]
   distribute-rgt-in [=>] 99.9%	\[ 1 - \color{blue}{\left(\left(x \cdot 0.12\right) \cdot x + 0.253 \cdot x\right)} \]
   fma-def [=>] 99.9%	\[ 1 - \color{blue}{\mathsf{fma}\left(x \cdot 0.12, x, 0.253 \cdot x\right)} \]
   *-commutative [=>] 99.9%	\[ 1 - \mathsf{fma}\left(x \cdot 0.12, x, \color{blue}{x \cdot 0.253}\right) \]

3. Applied egg-rr 66.2%

   \[\leadsto 1 - \color{blue}{\log \left(\frac{{\left(e^{x}\right)}^{0.253}}{{\left({\left(e^{-0.12}\right)}^{x}\right)}^{x}}\right)} \]
   Step-by-step derivation

   [Start] 99.9%	\[ 1 - \mathsf{fma}\left(x \cdot 0.12, x, x \cdot 0.253\right) \]
   fma-udef [=>] 99.9%	\[ 1 - \color{blue}{\left(\left(x \cdot 0.12\right) \cdot x + x \cdot 0.253\right)} \]
   *-commutative [=>] 99.9%	\[ 1 - \left(\color{blue}{\left(0.12 \cdot x\right)} \cdot x + x \cdot 0.253\right) \]
   associate-*r* [<=] 99.8%	\[ 1 - \left(\color{blue}{0.12 \cdot \left(x \cdot x\right)} + x \cdot 0.253\right) \]
   +-commutative [<=] 99.8%	\[ 1 - \color{blue}{\left(x \cdot 0.253 + 0.12 \cdot \left(x \cdot x\right)\right)} \]
   *-commutative [=>] 99.8%	\[ 1 - \left(\color{blue}{0.253 \cdot x} + 0.12 \cdot \left(x \cdot x\right)\right) \]
   associate-*r* [=>] 99.9%	\[ 1 - \left(0.253 \cdot x + \color{blue}{\left(0.12 \cdot x\right) \cdot x}\right) \]
   *-commutative [<=] 99.9%	\[ 1 - \left(0.253 \cdot x + \color{blue}{\left(x \cdot 0.12\right)} \cdot x\right) \]
   distribute-rgt-out [=>] 99.9%	\[ 1 - \color{blue}{x \cdot \left(0.253 + x \cdot 0.12\right)} \]
   *-commutative [=>] 99.9%	\[ 1 - x \cdot \left(0.253 + \color{blue}{0.12 \cdot x}\right) \]
   metadata-eval [<=] 99.9%	\[ 1 - x \cdot \left(0.253 + \color{blue}{\left(--0.12\right)} \cdot x\right) \]
   cancel-sign-sub-inv [<=] 99.9%	\[ 1 - x \cdot \color{blue}{\left(0.253 - -0.12 \cdot x\right)} \]
   add-sqr-sqrt [=>] 51.0%	\[ 1 - x \cdot \left(0.253 - \color{blue}{\sqrt{-0.12 \cdot x} \cdot \sqrt{-0.12 \cdot x}}\right) \]
   sqrt-unprod [=>] 76.7%	\[ 1 - x \cdot \left(0.253 - \color{blue}{\sqrt{\left(-0.12 \cdot x\right) \cdot \left(-0.12 \cdot x\right)}}\right) \]
   swap-sqr [=>] 76.7%	\[ 1 - x \cdot \left(0.253 - \sqrt{\color{blue}{\left(-0.12 \cdot -0.12\right) \cdot \left(x \cdot x\right)}}\right) \]
   metadata-eval [=>] 76.7%	\[ 1 - x \cdot \left(0.253 - \sqrt{\color{blue}{0.0144} \cdot \left(x \cdot x\right)}\right) \]
   *-commutative [<=] 76.7%	\[ 1 - x \cdot \left(0.253 - \sqrt{\color{blue}{\left(x \cdot x\right) \cdot 0.0144}}\right) \]
   metadata-eval [<=] 76.7%	\[ 1 - x \cdot \left(0.253 - \sqrt{\left(x \cdot x\right) \cdot \color{blue}{\left(0.12 \cdot 0.12\right)}}\right) \]
   swap-sqr [<=] 76.7%	\[ 1 - x \cdot \left(0.253 - \sqrt{\color{blue}{\left(x \cdot 0.12\right) \cdot \left(x \cdot 0.12\right)}}\right) \]
   sqrt-unprod [<=] 25.6%	\[ 1 - x \cdot \left(0.253 - \color{blue}{\sqrt{x \cdot 0.12} \cdot \sqrt{x \cdot 0.12}}\right) \]
   add-sqr-sqrt [<=] 53.7%	\[ 1 - x \cdot \left(0.253 - \color{blue}{x \cdot 0.12}\right) \]
   distribute-rgt-out-- [<=] 53.7%	\[ 1 - \color{blue}{\left(0.253 \cdot x - \left(x \cdot 0.12\right) \cdot x\right)} \]
   *-commutative [<=] 53.7%	\[ 1 - \left(\color{blue}{x \cdot 0.253} - \left(x \cdot 0.12\right) \cdot x\right) \]
   add-log-exp [=>] 54.2%	\[ 1 - \left(\color{blue}{\log \left(e^{x \cdot 0.253}\right)} - \left(x \cdot 0.12\right) \cdot x\right) \]

4. Simplified 99.9%

   \[\leadsto 1 - \color{blue}{\left(0.253 \cdot x - x \cdot \left(x \cdot -0.12\right)\right)} \]
   Step-by-step derivation

   [Start] 66.2%	\[ 1 - \log \left(\frac{{\left(e^{x}\right)}^{0.253}}{{\left({\left(e^{-0.12}\right)}^{x}\right)}^{x}}\right) \]
   log-div [=>] 66.2%	\[ 1 - \color{blue}{\left(\log \left({\left(e^{x}\right)}^{0.253}\right) - \log \left({\left({\left(e^{-0.12}\right)}^{x}\right)}^{x}\right)\right)} \]
   log-pow [=>] 66.2%	\[ 1 - \left(\color{blue}{0.253 \cdot \log \left(e^{x}\right)} - \log \left({\left({\left(e^{-0.12}\right)}^{x}\right)}^{x}\right)\right) \]
   fma-neg [=>] 66.2%	\[ 1 - \color{blue}{\mathsf{fma}\left(0.253, \log \left(e^{x}\right), -\log \left({\left({\left(e^{-0.12}\right)}^{x}\right)}^{x}\right)\right)} \]
   rem-log-exp [=>] 76.4%	\[ 1 - \mathsf{fma}\left(0.253, \color{blue}{x}, -\log \left({\left({\left(e^{-0.12}\right)}^{x}\right)}^{x}\right)\right) \]
   log-pow [=>] 76.4%	\[ 1 - \mathsf{fma}\left(0.253, x, -\color{blue}{x \cdot \log \left({\left(e^{-0.12}\right)}^{x}\right)}\right) \]
   log-pow [=>] 99.4%	\[ 1 - \mathsf{fma}\left(0.253, x, -x \cdot \color{blue}{\left(x \cdot \log \left(e^{-0.12}\right)\right)}\right) \]
   rem-log-exp [=>] 99.9%	\[ 1 - \mathsf{fma}\left(0.253, x, -x \cdot \left(x \cdot \color{blue}{-0.12}\right)\right) \]
   metadata-eval [<=] 99.9%	\[ 1 - \mathsf{fma}\left(0.253, x, -x \cdot \left(x \cdot \color{blue}{\left(-0.12\right)}\right)\right) \]
   distribute-rgt-neg-in [<=] 99.9%	\[ 1 - \mathsf{fma}\left(0.253, x, -x \cdot \color{blue}{\left(-x \cdot 0.12\right)}\right) \]
   distribute-rgt-neg-out [=>] 99.9%	\[ 1 - \mathsf{fma}\left(0.253, x, -\color{blue}{\left(-x \cdot \left(x \cdot 0.12\right)\right)}\right) \]
   associate-*r* [=>] 99.8%	\[ 1 - \mathsf{fma}\left(0.253, x, -\left(-\color{blue}{\left(x \cdot x\right) \cdot 0.12}\right)\right) \]
   *-commutative [<=] 99.8%	\[ 1 - \mathsf{fma}\left(0.253, x, -\left(-\color{blue}{0.12 \cdot \left(x \cdot x\right)}\right)\right) \]
   fma-neg [<=] 99.8%	\[ 1 - \color{blue}{\left(0.253 \cdot x - \left(-0.12 \cdot \left(x \cdot x\right)\right)\right)} \]
   distribute-lft-neg-in [=>] 99.8%	\[ 1 - \left(0.253 \cdot x - \color{blue}{\left(-0.12\right) \cdot \left(x \cdot x\right)}\right) \]
   metadata-eval [=>] 99.8%	\[ 1 - \left(0.253 \cdot x - \color{blue}{-0.12} \cdot \left(x \cdot x\right)\right) \]
   *-commutative [=>] 99.8%	\[ 1 - \left(0.253 \cdot x - \color{blue}{\left(x \cdot x\right) \cdot -0.12}\right) \]
   associate-*r* [<=] 99.9%	\[ 1 - \left(0.253 \cdot x - \color{blue}{x \cdot \left(x \cdot -0.12\right)}\right) \]

5. Final simplification 99.9%

   \[\leadsto 1 + \left(x \cdot \left(x \cdot -0.12\right) - 0.253 \cdot x\right) \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	704

\[1 + \left(x \cdot \left(x \cdot -0.12\right) - 0.253 \cdot x\right) \]

Alternative 2
Accuracy	59.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1.5334083333333333\\ \end{array} \]

Alternative 3
Accuracy	98.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - 0.253 \cdot x\\ \end{array} \]

Alternative 4
Accuracy	99.9%
Cost	576

\[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

Alternative 5
Accuracy	97.8%
Cost	448

\[1 - 0.12 \cdot \left(x \cdot x\right) \]

Alternative 6
Accuracy	97.8%
Cost	448

\[1 - x \cdot \left(x \cdot 0.12\right) \]

Alternative 7
Accuracy	10.5%
Cost	64

\[1.5334083333333333 \]

Reproduce

herbie shell --seed 2023165 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))