Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B

Average Accuracy: 99.9% → 99.9%

Time: 11.3s

Precision: binary64

Cost: 7488

Math

\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

↓

\[\mathsf{fma}\left(x, -0.70711, \frac{-1.6316775383 - x \cdot 0.1913510371}{-1 + x \cdot \left(x \cdot -0.04481 + -0.99229\right)}\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))

↓

(FPCore (x)
 :precision binary64
 (fma
  x
  -0.70711
  (/
   (- -1.6316775383 (* x 0.1913510371))
   (+ -1.0 (* x (+ (* x -0.04481) -0.99229))))))

C

double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

↓

double code(double x) {
	return fma(x, -0.70711, ((-1.6316775383 - (x * 0.1913510371)) / (-1.0 + (x * ((x * -0.04481) + -0.99229)))));
}

Julia

function code(x)
	return Float64(0.70711 * 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 fma(x, -0.70711, Float64(Float64(-1.6316775383 - Float64(x * 0.1913510371)) / Float64(-1.0 + Float64(x * Float64(Float64(x * -0.04481) + -0.99229)))))
end

Wolfram

code[x_] := N[(0.70711 * 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]), $MachinePrecision]

↓

code[x_] := N[(x * -0.70711 + N[(N[(-1.6316775383 - N[(x * 0.1913510371), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(x * N[(N[(x * -0.04481), $MachinePrecision] + -0.99229), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

2. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)} \]
   Proof

   [Start] 99.9	\[ 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   sub-neg [=>] 99.9	\[ 0.70711 \cdot \color{blue}{\left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} + \left(-x\right)\right)} \]
   distribute-lft-in [=>] 99.9	\[ \color{blue}{0.70711 \cdot \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} + 0.70711 \cdot \left(-x\right)} \]
   +-commutative [=>] 99.9	\[ \color{blue}{0.70711 \cdot \left(-x\right) + 0.70711 \cdot \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} \]
   neg-mul-1 [=>] 99.9	\[ 0.70711 \cdot \color{blue}{\left(-1 \cdot x\right)} + 0.70711 \cdot \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \]
   associate-*r* [=>] 99.9	\[ \color{blue}{\left(0.70711 \cdot -1\right) \cdot x} + 0.70711 \cdot \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \]
   *-commutative [=>] 99.9	\[ \color{blue}{x \cdot \left(0.70711 \cdot -1\right)} + 0.70711 \cdot \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \]
   *-commutative [<=] 99.9	\[ x \cdot \left(0.70711 \cdot -1\right) + \color{blue}{\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \cdot 0.70711} \]
   fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(x, 0.70711 \cdot -1, \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \cdot 0.70711\right)} \]
   metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(x, \color{blue}{-0.70711}, \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \cdot 0.70711\right) \]
   associate-*l/ [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \color{blue}{\frac{\left(2.30753 + x \cdot 0.27061\right) \cdot 0.70711}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}}\right) \]
   *-commutative [<=] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{\color{blue}{0.70711 \cdot \left(2.30753 + x \cdot 0.27061\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]
   +-commutative [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{0.70711 \cdot \color{blue}{\left(x \cdot 0.27061 + 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]
   distribute-lft-in [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{\color{blue}{0.70711 \cdot \left(x \cdot 0.27061\right) + 0.70711 \cdot 2.30753}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]
   associate-*r* [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{\color{blue}{\left(0.70711 \cdot x\right) \cdot 0.27061} + 0.70711 \cdot 2.30753}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]
   *-commutative [<=] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{\color{blue}{\left(x \cdot 0.70711\right)} \cdot 0.27061 + 0.70711 \cdot 2.30753}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]
   associate-*l* [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{\color{blue}{x \cdot \left(0.70711 \cdot 0.27061\right)} + 0.70711 \cdot 2.30753}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]
   fma-def [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{\color{blue}{\mathsf{fma}\left(x, 0.70711 \cdot 0.27061, 0.70711 \cdot 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]
   metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]
   metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, \color{blue}{1.6316775383}\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

3. Applied egg-rr 99.9%

   \[\leadsto \mathsf{fma}\left(x, -0.70711, \color{blue}{-\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{-1 - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}}\right) \]
   Proof

   [Start] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right) \]
   frac-2neg [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \color{blue}{\frac{-\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{-\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}\right) \]
   distribute-frac-neg [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \color{blue}{-\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{-\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}\right) \]
   neg-sub0 [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, -\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{0 - \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}\right) \]
   metadata-eval [<=] 99.9	\[ \mathsf{fma}\left(x, -0.70711, -\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{\log 1} - \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right) \]
   fma-udef [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, -\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\log 1 - \color{blue}{\left(x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right) + 1\right)}}\right) \]
   +-commutative [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, -\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\log 1 - \color{blue}{\left(1 + x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)\right)}}\right) \]
   associate--r+ [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, -\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{\left(\log 1 - 1\right) - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}}\right) \]
   metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, -\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\left(\color{blue}{0} - 1\right) - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}\right) \]
   metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, -\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{-1} - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}\right) \]

4. Simplified 99.9%

   \[\leadsto \mathsf{fma}\left(x, -0.70711, \color{blue}{\frac{-1.6316775383 - x \cdot 0.1913510371}{-1 - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}}\right) \]
   Proof

   [Start] 99.9	\[ \mathsf{fma}\left(x, -0.70711, -\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{-1 - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}\right) \]
   distribute-neg-frac [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \color{blue}{\frac{-\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{-1 - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}}\right) \]
   fma-def [<=] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{-\color{blue}{\left(x \cdot 0.1913510371 + 1.6316775383\right)}}{-1 - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}\right) \]
   +-commutative [<=] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{-\color{blue}{\left(1.6316775383 + x \cdot 0.1913510371\right)}}{-1 - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}\right) \]
   distribute-neg-in [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{\color{blue}{\left(-1.6316775383\right) + \left(-x \cdot 0.1913510371\right)}}{-1 - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}\right) \]
   metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{\color{blue}{-1.6316775383} + \left(-x \cdot 0.1913510371\right)}{-1 - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}\right) \]
   sub-neg [<=] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{\color{blue}{-1.6316775383 - x \cdot 0.1913510371}}{-1 - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}\right) \]

5. Applied egg-rr 99.9%

   \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{-1.6316775383 - x \cdot 0.1913510371}{-1 - x \cdot \color{blue}{\left(x \cdot 0.04481 + 0.99229\right)}}\right) \]
   Proof

   [Start] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{-1.6316775383 - x \cdot 0.1913510371}{-1 - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}\right) \]
   fma-udef [=>] 99.9	\[ \mathsf{fma}\left(x, -0.70711, \frac{-1.6316775383 - x \cdot 0.1913510371}{-1 - x \cdot \color{blue}{\left(x \cdot 0.04481 + 0.99229\right)}}\right) \]

6. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{-1.6316775383 - x \cdot 0.1913510371}{-1 + x \cdot \left(x \cdot -0.04481 + -0.99229\right)}\right) \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	1216

\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(x \cdot 0.04481 + 0.99229\right)} - x\right) \]

Alternative 2
Accuracy	98.4%
Cost	960

\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot 0.99229} - x\right) \]

Alternative 3
Accuracy	98.1%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 4
Accuracy	97.7%
Cost	320

\[0.70711 \cdot \left(2.30753 - x\right) \]

Alternative 5
Accuracy	97.7%
Cost	320

\[\frac{2.30753 - x}{1.4142071247754946} \]

Alternative 6
Accuracy	49.9%
Cost	64

\[1.6316775383 \]

Reproduce

herbie shell --seed 2023140 
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))