Average Error: 21.5 → 0.3
Time: 11.7s
Precision: 64
Internal Precision: 832
\[x \cdot \left(1 - \sqrt{1 - 0.6 \cdot x}\right)\]
\[\begin{array}{l} \mathbf{if}\;x \le -0.00020712147214625587:\\ \;\;\;\;x + \sqrt{1 - 0.6 \cdot x} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.0135 \cdot \left(x \cdot x\right) + \left(x \cdot 0.045 + 0.3\right)\right) \cdot x\right) \cdot x\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if x < -0.00020712147214625587

    1. Initial program 0.4

      \[x \cdot \left(1 - \sqrt{1 - 0.6 \cdot x}\right)\]
    2. Using strategy rm
    3. Applied sub-neg0.4

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(-\sqrt{1 - 0.6 \cdot x}\right)\right)}\]
    4. Applied distribute-lft-in0.4

      \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left(-\sqrt{1 - 0.6 \cdot x}\right)}\]

    if -0.00020712147214625587 < x

    1. Initial program 28.8

      \[x \cdot \left(1 - \sqrt{1 - 0.6 \cdot x}\right)\]
    2. Taylor expanded around 0 0.3

      \[\leadsto x \cdot \color{blue}{\left(0.0135 \cdot {x}^{3} + \left(0.045 \cdot {x}^{2} + 0.3 \cdot x\right)\right)}\]
    3. Simplified0.3

      \[\leadsto x \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot 0.0135 + \left(0.3 + 0.045 \cdot x\right)\right) \cdot x\right)}\]
  3. Recombined 2 regimes into one program.
  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00020712147214625587:\\ \;\;\;\;x + \sqrt{1 - 0.6 \cdot x} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.0135 \cdot \left(x \cdot x\right) + \left(x \cdot 0.045 + 0.3\right)\right) \cdot x\right) \cdot x\\ \end{array}\]

Runtime

Time bar (total: 11.7s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x)
  :name "x * (1 - sqrt(1-0.6*x))"
  (* x (- 1 (sqrt (- 1 (* 0.6 x))))))