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

Average Error: 0.17% → 0.16%

Time: 4.8s

Precision: binary64

Cost: 6848

Math

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

↓

\[1 - x \cdot \mathsf{fma}\left(x, 0.12, 0.253\right) \]

FPCore

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

↓

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

C

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

↓

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

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(x * fma(x, 0.12, 0.253)))
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[(x * N[(x * 0.12 + 0.253), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.17

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

2. Simplified 0.16

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

   [Start] 0.17	\[ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   distribute-lft-in [=>] 0.17	\[ 1 - \color{blue}{\left(x \cdot 0.253 + x \cdot \left(x \cdot 0.12\right)\right)} \]
   +-commutative [=>] 0.17	\[ 1 - \color{blue}{\left(x \cdot \left(x \cdot 0.12\right) + x \cdot 0.253\right)} \]
   *-commutative [<=] 0.17	\[ 1 - \left(\color{blue}{\left(x \cdot 0.12\right) \cdot x} + x \cdot 0.253\right) \]
   cancel-sign-sub [<=] 0.17	\[ 1 - \color{blue}{\left(\left(x \cdot 0.12\right) \cdot x - \left(-x\right) \cdot 0.253\right)} \]
   *-commutative [<=] 0.17	\[ 1 - \left(\left(x \cdot 0.12\right) \cdot x - \color{blue}{0.253 \cdot \left(-x\right)}\right) \]
   *-commutative [=>] 0.17	\[ 1 - \left(\left(x \cdot 0.12\right) \cdot x - \color{blue}{\left(-x\right) \cdot 0.253}\right) \]
   cancel-sign-sub [=>] 0.17	\[ 1 - \color{blue}{\left(\left(x \cdot 0.12\right) \cdot x + x \cdot 0.253\right)} \]
   *-commutative [=>] 0.17	\[ 1 - \left(\color{blue}{x \cdot \left(x \cdot 0.12\right)} + x \cdot 0.253\right) \]
   distribute-lft-in [<=] 0.17	\[ 1 - \color{blue}{x \cdot \left(x \cdot 0.12 + 0.253\right)} \]
   fma-def [=>] 0.16	\[ 1 - x \cdot \color{blue}{\mathsf{fma}\left(x, 0.12, 0.253\right)} \]

3. Final simplification 0.16

   \[\leadsto 1 - x \cdot \mathsf{fma}\left(x, 0.12, 0.253\right) \]

Alternatives

Alternative 1
Error	3.1%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \lor \neg \left(x \leq 2.1\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	2.39%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \lor \neg \left(x \leq 2.1\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot -0.253\\ \end{array} \]

Alternative 3
Error	0.17%
Cost	576

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

Alternative 4
Error	3.11%
Cost	448

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

Alternative 5
Error	33.96%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))