Average Error: 16.8 → 15.5
Time: 4.9s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le 2.5714192298972582 \cdot 10^{-69}:\\ \;\;\;\;\frac{x + 1 \cdot \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le 4.90312964716450283 \cdot 10^{254}:\\ \;\;\;\;\sqrt{\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}} \cdot \sqrt{\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{\frac{\frac{t}{z}}{y}}\right) \cdot \frac{1}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le 2.5714192298972582 \cdot 10^{-69}:\\
\;\;\;\;\frac{x + 1 \cdot \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\

\mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le 4.90312964716450283 \cdot 10^{254}:\\
\;\;\;\;\sqrt{\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}} \cdot \sqrt{\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r699490 = x;
        double r699491 = y;
        double r699492 = z;
        double r699493 = r699491 * r699492;
        double r699494 = t;
        double r699495 = r699493 / r699494;
        double r699496 = r699490 + r699495;
        double r699497 = a;
        double r699498 = 1.0;
        double r699499 = r699497 + r699498;
        double r699500 = b;
        double r699501 = r699491 * r699500;
        double r699502 = r699501 / r699494;
        double r699503 = r699499 + r699502;
        double r699504 = r699496 / r699503;
        return r699504;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r699505 = x;
        double r699506 = y;
        double r699507 = z;
        double r699508 = r699506 * r699507;
        double r699509 = t;
        double r699510 = r699508 / r699509;
        double r699511 = r699505 + r699510;
        double r699512 = a;
        double r699513 = 1.0;
        double r699514 = r699512 + r699513;
        double r699515 = b;
        double r699516 = r699506 * r699515;
        double r699517 = r699516 / r699509;
        double r699518 = r699514 + r699517;
        double r699519 = r699511 / r699518;
        double r699520 = 2.571419229897258e-69;
        bool r699521 = r699519 <= r699520;
        double r699522 = 1.0;
        double r699523 = r699509 / r699507;
        double r699524 = r699506 / r699523;
        double r699525 = r699522 * r699524;
        double r699526 = r699505 + r699525;
        double r699527 = r699515 / r699509;
        double r699528 = r699506 * r699527;
        double r699529 = r699514 + r699528;
        double r699530 = r699526 / r699529;
        double r699531 = 4.903129647164503e+254;
        bool r699532 = r699519 <= r699531;
        double r699533 = sqrt(r699519);
        double r699534 = r699533 * r699533;
        double r699535 = r699523 / r699506;
        double r699536 = r699522 / r699535;
        double r699537 = r699505 + r699536;
        double r699538 = r699522 / r699529;
        double r699539 = r699537 * r699538;
        double r699540 = r699532 ? r699534 : r699539;
        double r699541 = r699521 ? r699530 : r699540;
        return r699541;
}

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

Original16.8
Target13.1
Herbie15.5
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < 2.571419229897258e-69

    1. Initial program 11.2

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

    3. Applied *-un-lft-identity11.2

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

    4. Applied times-frac11.2

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

    5. Simplified11.2

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

    7. Applied associate-/l*11.4

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

    9. Applied clear-num11.4

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

    11. Applied *-un-lft-identity11.4

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

    12. Applied *-un-lft-identity11.4

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

    13. Applied *-un-lft-identity11.4

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

    14. Applied times-frac11.4

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

    15. Applied times-frac11.4

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

    16. Applied add-sqr-sqrt11.4

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

    17. Applied times-frac11.4

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

    18. Simplified11.4

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

    19. Simplified11.4

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

    if 2.571419229897258e-69 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) < 4.903129647164503e+254

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt0.6

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

    if 4.903129647164503e+254 < (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))

    1. Initial program 58.9

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

    3. Applied *-un-lft-identity58.9

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

    4. Applied times-frac58.9

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

    5. Simplified58.9

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

    7. Applied associate-/l*49.6

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

    9. Applied clear-num49.6

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

    11. Applied div-inv49.7

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le 2.5714192298972582 \cdot 10^{-69}:\\ \;\;\;\;\frac{x + 1 \cdot \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \le 4.90312964716450283 \cdot 10^{254}:\\ \;\;\;\;\sqrt{\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}} \cdot \sqrt{\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{\frac{\frac{t}{z}}{y}}\right) \cdot \frac{1}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

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