Average Error: 20.4 → 5.5
Time: 17.1s
Precision: 64
Internal Precision: 128
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.3240045460113365 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.2941982648075124 \cdot 10^{-160} \lor \neg \left(y \le 8.707755780430149 \cdot 10^{-162}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.4
Target0.1
Herbie5.5
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.3240045460113365e+154

    1. Initial program 63.6

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]

    2. Initial simplification63.6

      \[\leadsto \frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\]

    3. Taylor expanded around 0 0

      \[\leadsto \color{blue}{-1}\]

    if -1.3240045460113365e+154 < y < -1.2941982648075124e-160 or 8.707755780430149e-162 < y

    1. Initial program 0.0

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]

    2. Initial simplification0.0

      \[\leadsto \frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\]

    if -1.2941982648075124e-160 < y < 8.707755780430149e-162

    1. Initial program 30.4

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]

    2. Initial simplification30.4

      \[\leadsto \frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt30.4

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

    5. Applied times-frac30.7

      \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{y + x}{\sqrt{x \cdot x + y \cdot y}}}\]

    6. Taylor expanded around inf 16.9

      \[\leadsto \color{blue}{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3240045460113365 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.2941982648075124 \cdot 10^{-160} \lor \neg \left(y \le 8.707755780430149 \cdot 10^{-162}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Runtime

Time bar (total: 17.1s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes21.55.50.620.976.4%
herbie shell --seed 2018352 
(FPCore (x y)
  :name "Kahan p9 Example"
  :pre (and (< 0 x 1) (< y 1))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))