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

Average Accuracy: 100.0% → 100.0%

Time: 8.5s

Precision: binary64

Cost: 13760

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

↓

function code(x)
	return Float64(Float64(Float64(-1.0 / fma(x, fma(x, 0.04481, 0.99229), 1.0)) * Float64(-2.30753 + Float64(x * -0.27061))) - x)
end

Wolfram

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

↓

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

Derivation

1. Initial program 100.0%

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

2. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   frac-2neg [=>] 100.0	\[ \color{blue}{\frac{-\left(2.30753 + x \cdot 0.27061\right)}{-\left(1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)\right)}} - x \]
   clear-num [=>] 100.0	\[ \color{blue}{\frac{1}{\frac{-\left(1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)\right)}{-\left(2.30753 + x \cdot 0.27061\right)}}} - x \]
   associate-/r/ [=>] 100.0	\[ \color{blue}{\frac{1}{-\left(1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)\right)} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right)} - x \]
   *-un-lft-identity [=>] 100.0	\[ \frac{1}{-\color{blue}{1 \cdot \left(1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)\right)}} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) - x \]
   distribute-lft-neg-in [=>] 100.0	\[ \frac{1}{\color{blue}{\left(-1\right) \cdot \left(1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)\right)}} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) - x \]
   associate-/r* [=>] 100.0	\[ \color{blue}{\frac{\frac{1}{-1}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) - x \]
   metadata-eval [=>] 100.0	\[ \frac{\frac{1}{\color{blue}{-1}}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) - x \]
   metadata-eval [=>] 100.0	\[ \frac{\color{blue}{-1}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) - x \]
   +-commutative [=>] 100.0	\[ \frac{-1}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) - x \]
   fma-def [=>] 100.0	\[ \frac{-1}{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) - x \]
   +-commutative [=>] 100.0	\[ \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.04481 + 0.99229}, 1\right)} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) - x \]
   fma-def [=>] 100.0	\[ \frac{-1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)}, 1\right)} \cdot \left(-\left(2.30753 + x \cdot 0.27061\right)\right) - x \]
   neg-sub0 [=>] 100.0	\[ \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \color{blue}{\left(0 - \left(2.30753 + x \cdot 0.27061\right)\right)} - x \]
   metadata-eval [<=] 100.0	\[ \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \left(\color{blue}{\log 1} - \left(2.30753 + x \cdot 0.27061\right)\right) - x \]
   associate--r+ [=>] 100.0	\[ \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \color{blue}{\left(\left(\log 1 - 2.30753\right) - x \cdot 0.27061\right)} - x \]
   metadata-eval [=>] 100.0	\[ \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \left(\left(\color{blue}{0} - 2.30753\right) - x \cdot 0.27061\right) - x \]
   metadata-eval [=>] 100.0	\[ \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \left(\color{blue}{-2.30753} - x \cdot 0.27061\right) - x \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	1088

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

Alternative 2
Accuracy	98.5%
Cost	832

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

Alternative 3
Accuracy	98.3%
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4
Accuracy	97.8%
Cost	192

\[2.30753 - x \]

Alternative 5
Accuracy	49.9%
Cost	64

\[2.30753 \]

Reproduce

herbie shell --seed 2023140 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))