Average Error: 33.8 → 16.9
Time: 27.6s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-x\right) - \sqrt{x \cdot x - \left(4 \cdot y\right) \cdot z}}{2 \cdot z}\]
\[\begin{array}{l} \mathbf{if}\;x \le -4.178732388010056 \cdot 10^{-128}:\\ \;\;\;\;\frac{4 \cdot \left(z \cdot y\right)}{\left(z \cdot 2\right) \cdot \left(\sqrt{x \cdot x - z \cdot \left(y \cdot 4\right)} + \left(-x\right)\right)}\\ \mathbf{elif}\;x \le 2.7512793042658205 \cdot 10^{+92}:\\ \;\;\;\;\frac{-x}{z \cdot 2} - \frac{\sqrt{x \cdot x - z \cdot \left(y \cdot 4\right)}}{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes
  2. if x < -4.178732388010056e-128

    1. Initial program 50.8

      \[\frac{\left(-x\right) - \sqrt{x \cdot x - \left(4 \cdot y\right) \cdot z}}{2 \cdot z}\]
    2. Initial simplification50.8

      \[\leadsto \frac{\left(-x\right) - \sqrt{x \cdot x - z \cdot \left(4 \cdot y\right)}}{2 \cdot z}\]
    3. Using strategy rm
    4. Applied flip--50.8

      \[\leadsto \frac{\color{blue}{\frac{\left(-x\right) \cdot \left(-x\right) - \sqrt{x \cdot x - z \cdot \left(4 \cdot y\right)} \cdot \sqrt{x \cdot x - z \cdot \left(4 \cdot y\right)}}{\left(-x\right) + \sqrt{x \cdot x - z \cdot \left(4 \cdot y\right)}}}}{2 \cdot z}\]
    5. Applied associate-/l/52.1

      \[\leadsto \color{blue}{\frac{\left(-x\right) \cdot \left(-x\right) - \sqrt{x \cdot x - z \cdot \left(4 \cdot y\right)} \cdot \sqrt{x \cdot x - z \cdot \left(4 \cdot y\right)}}{\left(2 \cdot z\right) \cdot \left(\left(-x\right) + \sqrt{x \cdot x - z \cdot \left(4 \cdot y\right)}\right)}}\]
    6. Simplified26.3

      \[\leadsto \frac{\color{blue}{\left(z \cdot y\right) \cdot 4}}{\left(2 \cdot z\right) \cdot \left(\left(-x\right) + \sqrt{x \cdot x - z \cdot \left(4 \cdot y\right)}\right)}\]

    if -4.178732388010056e-128 < x < 2.7512793042658205e+92

    1. Initial program 11.9

      \[\frac{\left(-x\right) - \sqrt{x \cdot x - \left(4 \cdot y\right) \cdot z}}{2 \cdot z}\]
    2. Initial simplification11.9

      \[\leadsto \frac{\left(-x\right) - \sqrt{x \cdot x - z \cdot \left(4 \cdot y\right)}}{2 \cdot z}\]
    3. Using strategy rm
    4. Applied div-sub11.9

      \[\leadsto \color{blue}{\frac{-x}{2 \cdot z} - \frac{\sqrt{x \cdot x - z \cdot \left(4 \cdot y\right)}}{2 \cdot z}}\]

    if 2.7512793042658205e+92 < x

    1. Initial program 42.6

      \[\frac{\left(-x\right) - \sqrt{x \cdot x - \left(4 \cdot y\right) \cdot z}}{2 \cdot z}\]
    2. Initial simplification42.6

      \[\leadsto \frac{\left(-x\right) - \sqrt{x \cdot x - z \cdot \left(4 \cdot y\right)}}{2 \cdot z}\]
    3. Taylor expanded around inf 4.2

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{z}}\]
    4. Simplified4.2

      \[\leadsto \color{blue}{-\frac{x}{z}}\]
  3. Recombined 3 regimes into one program.
  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.178732388010056 \cdot 10^{-128}:\\ \;\;\;\;\frac{4 \cdot \left(z \cdot y\right)}{\left(z \cdot 2\right) \cdot \left(\sqrt{x \cdot x - z \cdot \left(y \cdot 4\right)} + \left(-x\right)\right)}\\ \mathbf{elif}\;x \le 2.7512793042658205 \cdot 10^{+92}:\\ \;\;\;\;\frac{-x}{z \cdot 2} - \frac{\sqrt{x \cdot x - z \cdot \left(y \cdot 4\right)}}{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z}\\ \end{array}\]

Runtime

Time bar (total: 27.6s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x y z)
  :name "(-x-sqrt(x*x-4*y*z))/(2z)"
  (/ (- (- x) (sqrt (- (* x x) (* (* 4 y) z)))) (* 2 z)))