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

Average Error: 0.02% → 0.02%

Time: 7.8s

Precision: binary64

Cost: 7360

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (- (/ (+ (* x 0.27061) 2.30753) (fma x (+ (* x 0.04481) 0.99229) 1.0)) 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 (((x * 0.27061) + 2.30753) / fma(x, ((x * 0.04481) + 0.99229), 1.0)) - 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(Float64(x * 0.27061) + 2.30753) / fma(x, Float64(Float64(x * 0.04481) + 0.99229), 1.0)) - 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[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(x * N[(N[(x * 0.04481), $MachinePrecision] + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 0.02

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

2. Simplified 0.02

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

   [Start] 0.02	\[ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   remove-double-neg [<=] 0.02	\[ \color{blue}{\left(-\left(-\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)\right)} - x \]
   neg-mul-1 [=>] 0.02	\[ \color{blue}{-1 \cdot \left(-\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)} - x \]
   *-commutative [=>] 0.02	\[ \color{blue}{\left(-\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \cdot -1} - x \]
   *-commutative [<=] 0.02	\[ \color{blue}{-1 \cdot \left(-\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)} - x \]
   neg-mul-1 [<=] 0.02	\[ \color{blue}{\left(-\left(-\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)\right)} - x \]
   remove-double-neg [=>] 0.02	\[ \color{blue}{\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} - x \]
   +-commutative [=>] 0.02	\[ \frac{\color{blue}{x \cdot 0.27061 + 2.30753}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   fma-def [=>] 0.02	\[ \frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   +-commutative [=>] 0.02	\[ \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}} - x \]
   fma-def [=>] 0.02	\[ \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}} - x \]
   +-commutative [=>] 0.02	\[ \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.04481 + 0.99229}, 1\right)} - x \]
   fma-def [=>] 0.02	\[ \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)}, 1\right)} - x \]

3. Applied egg-rr 0.02

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

4. Applied egg-rr 0.02

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

5. Final simplification 0.02

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

Alternatives

Alternative 1
Error	0.02%
Cost	1088

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

Alternative 2
Error	1.16%
Cost	832

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

Alternative 3
Error	1.49%
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4
Error	1.9%
Cost	192

\[2.30753 - x \]

Alternative 5
Error	48.87%
Cost	64

\[2.30753 \]

Reproduce

herbie shell --seed 2023121 
(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))