Average Error: 16.6 → 12.9
Time: 13.9s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.591884767125234007189270192734521744478 \cdot 10^{-16}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{1}{\frac{\frac{t}{y}}{b}}}\\ \mathbf{elif}\;t \le 3.078693643875039956031098367503131499945 \cdot 10^{59}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{1}{t}}{\frac{1}{y \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{1}{\frac{\frac{t}{y}}{b}}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -3.591884767125234007189270192734521744478 \cdot 10^{-16}:\\
\;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{1}{\frac{\frac{t}{y}}{b}}}\\

\mathbf{elif}\;t \le 3.078693643875039956031098367503131499945 \cdot 10^{59}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{1}{t}}{\frac{1}{y \cdot b}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{1}{\frac{\frac{t}{y}}{b}}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r565327 = x;
        double r565328 = y;
        double r565329 = z;
        double r565330 = r565328 * r565329;
        double r565331 = t;
        double r565332 = r565330 / r565331;
        double r565333 = r565327 + r565332;
        double r565334 = a;
        double r565335 = 1.0;
        double r565336 = r565334 + r565335;
        double r565337 = b;
        double r565338 = r565328 * r565337;
        double r565339 = r565338 / r565331;
        double r565340 = r565336 + r565339;
        double r565341 = r565333 / r565340;
        return r565341;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r565342 = t;
        double r565343 = -3.591884767125234e-16;
        bool r565344 = r565342 <= r565343;
        double r565345 = x;
        double r565346 = y;
        double r565347 = z;
        double r565348 = r565347 / r565342;
        double r565349 = r565346 * r565348;
        double r565350 = r565345 + r565349;
        double r565351 = a;
        double r565352 = 1.0;
        double r565353 = r565351 + r565352;
        double r565354 = 1.0;
        double r565355 = r565342 / r565346;
        double r565356 = b;
        double r565357 = r565355 / r565356;
        double r565358 = r565354 / r565357;
        double r565359 = r565353 + r565358;
        double r565360 = r565350 / r565359;
        double r565361 = 3.07869364387504e+59;
        bool r565362 = r565342 <= r565361;
        double r565363 = r565346 * r565347;
        double r565364 = r565363 / r565342;
        double r565365 = r565345 + r565364;
        double r565366 = r565354 / r565342;
        double r565367 = r565346 * r565356;
        double r565368 = r565354 / r565367;
        double r565369 = r565366 / r565368;
        double r565370 = r565353 + r565369;
        double r565371 = r565365 / r565370;
        double r565372 = r565342 / r565347;
        double r565373 = r565346 / r565372;
        double r565374 = r565345 + r565373;
        double r565375 = r565374 / r565359;
        double r565376 = r565362 ? r565371 : r565375;
        double r565377 = r565344 ? r565360 : r565376;
        return r565377;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.6
Target13.5
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -3.591884767125234e-16

    1. Initial program 12.3

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied clear-num12.3

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \color{blue}{\frac{1}{\frac{t}{y \cdot b}}}}\]
    4. Using strategy rm

    5. Applied associate-/r*9.6

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{1}{\color{blue}{\frac{\frac{t}{y}}{b}}}}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity9.6

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

    8. Applied times-frac5.0

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

    9. Simplified5.0

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

    if -3.591884767125234e-16 < t < 3.07869364387504e+59

    1. Initial program 20.6

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied clear-num20.7

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \color{blue}{\frac{1}{\frac{t}{y \cdot b}}}}\]
    4. Using strategy rm

    5. Applied div-inv20.7

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

    6. Applied associate-/r*20.7

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

    if 3.07869364387504e+59 < t

    1. Initial program 12.1

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied clear-num12.1

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \color{blue}{\frac{1}{\frac{t}{y \cdot b}}}}\]
    4. Using strategy rm

    5. Applied associate-/r*8.5

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{1}{\color{blue}{\frac{\frac{t}{y}}{b}}}}\]
    6. Using strategy rm

    7. Applied associate-/l*3.2

      \[\leadsto \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{1}{\frac{\frac{t}{y}}{b}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.591884767125234007189270192734521744478 \cdot 10^{-16}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{1}{\frac{\frac{t}{y}}{b}}}\\ \mathbf{elif}\;t \le 3.078693643875039956031098367503131499945 \cdot 10^{59}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{1}{t}}{\frac{1}{y \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{1}{\frac{\frac{t}{y}}{b}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 3.0369671037372459e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1) (/ (* y b) t))))