Average Error: 0.0 → 0.0

Time: 3.4s

Precision: binary64

Cost: 14080

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

↓

\[0.70711 \cdot \left(\sqrt[3]{{\left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)}^{3}} - x\right)\]

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

↓

0.70711 \cdot \left(\sqrt[3]{{\left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)}^{3}} - x\right)

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

↓

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (-
   (cbrt
    (pow
     (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
     3.0))
   x)))

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 0.70711 * (cbrt(pow(((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))), 3.0)) - x);
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	0.0
Cost	1216

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

Alternative 2
Error	0.9
Cost	960

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

Alternative 3
Error	1.4
Cost	320

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

Alternative 4
Error	0.7
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0784931178144082 \lor \neg \left(x \leq 1.1363965878489581\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 - x \cdot 2.134856267379707\\ \end{array}\]

Alternative 5
Error	1.1
Cost	520

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0784931178144082 \lor \neg \left(x \leq 1.17885225853273\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array}\]

Alternative 6
Error	31.8
Cost	64

\[1.6316775383\]

Alternative 7
Error	56.7
Cost	64

\[1\]

Derivation

Initial program 0.0
\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)\]
Using strategy rm
Applied add-cbrt-cube_binary64_35240.0
\[\leadsto 0.70711 \cdot \left(\color{blue}{\sqrt[3]{\left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \cdot \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \cdot \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}}} - x\right)\]
Simplified0.0
\[\leadsto 0.70711 \cdot \left(\sqrt[3]{\color{blue}{{\left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)}^{3}}} - x\right)\]
Simplified0.0
\[\leadsto \color{blue}{0.70711 \cdot \left(\sqrt[3]{{\left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)}^{3}} - x\right)}\]
Final simplification0.0
\[\leadsto 0.70711 \cdot \left(\sqrt[3]{{\left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)}^{3}} - x\right)\]

Reproduce

herbie shell --seed 2021044 
(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)))