Average Error: 16.0 → 15.7
Time: 1.4m
Precision: 64
Internal Precision: 384
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.1703200330472533 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{x + y \cdot \frac{z}{t}}{a - 1.0}}{\left(1.0 + a\right) \cdot \frac{t}{b} + y} \cdot \left(\left(a - 1.0\right) \cdot \frac{t}{b}\right)\\ \mathbf{if}\;y \le 4.872599708992765 \cdot 10^{+86}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + y \cdot \frac{z}{t}}{a - 1.0}}{\left(1.0 + a\right) \cdot \frac{t}{b} + y} \cdot \left(\left(a - 1.0\right) \cdot \frac{t}{b}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original16.0
Target13.1
Herbie15.7
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{if}\;t \lt 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.1703200330472533e+174 or 4.872599708992765e+86 < y

    1. Initial program 35.3

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied div-inv35.3

      \[\leadsto \frac{x + \color{blue}{\left(y \cdot z\right) \cdot \frac{1}{t}}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]

    4. Taylor expanded around 0 35.3

      \[\leadsto \frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1.0\right) + \color{blue}{\frac{y \cdot b}{t}}}\]

    5. Applied simplify29.9

      \[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1.0\right) + \frac{y}{\frac{t}{b}}}}\]
    6. Using strategy rm

    7. Applied flip-+36.5

      \[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\color{blue}{\frac{a \cdot a - 1.0 \cdot 1.0}{a - 1.0}} + \frac{y}{\frac{t}{b}}}\]

    8. Applied frac-add46.7

      \[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\color{blue}{\frac{\left(a \cdot a - 1.0 \cdot 1.0\right) \cdot \frac{t}{b} + \left(a - 1.0\right) \cdot y}{\left(a - 1.0\right) \cdot \frac{t}{b}}}}\]

    9. Applied associate-/r/45.4

      \[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a \cdot a - 1.0 \cdot 1.0\right) \cdot \frac{t}{b} + \left(a - 1.0\right) \cdot y} \cdot \left(\left(a - 1.0\right) \cdot \frac{t}{b}\right)}\]

    10. Applied simplify34.9

      \[\leadsto \color{blue}{\frac{\frac{x + y \cdot \frac{z}{t}}{a - 1.0}}{\left(1.0 + a\right) \cdot \frac{t}{b} + y}} \cdot \left(\left(a - 1.0\right) \cdot \frac{t}{b}\right)\]

    if -1.1703200330472533e+174 < y < 4.872599708992765e+86

    1. Initial program 8.5

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied div-inv8.5

      \[\leadsto \frac{x + \color{blue}{\left(y \cdot z\right) \cdot \frac{1}{t}}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]

    4. Taylor expanded around 0 8.5

      \[\leadsto \frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1.0\right) + \color{blue}{\frac{y \cdot b}{t}}}\]

    5. Applied simplify10.4

      \[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1.0\right) + \frac{y}{\frac{t}{b}}}}\]

    6. Taylor expanded around 0 8.3

      \[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1.0\right) + \color{blue}{\frac{y \cdot b}{t}}}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.4m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))