Average Error: 16.7 → 14.7
Time: 16.1s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.96219683641601833436006384613172315374 \cdot 10^{-210}:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{\frac{y}{t}} \cdot \sqrt[3]{\frac{y}{t}}\right) \cdot \left(z \cdot \sqrt[3]{\frac{y}{t}}\right)}{\left(b \cdot \frac{y}{t} + a\right) + 1}\\ \mathbf{elif}\;z \le 2.123087533092711734733727601124115985261 \cdot 10^{-209}:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{\sqrt[3]{y} \cdot z}{t}}{\left(a + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{b \cdot \sqrt[3]{y}}{t}\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(b \cdot \frac{y}{t} + a\right) + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.96219683641601833436006384613172315374 \cdot 10^{-210}:\\
\;\;\;\;\frac{x + \left(\sqrt[3]{\frac{y}{t}} \cdot \sqrt[3]{\frac{y}{t}}\right) \cdot \left(z \cdot \sqrt[3]{\frac{y}{t}}\right)}{\left(b \cdot \frac{y}{t} + a\right) + 1}\\

\mathbf{elif}\;z \le 2.123087533092711734733727601124115985261 \cdot 10^{-209}:\\
\;\;\;\;\frac{x + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{\sqrt[3]{y} \cdot z}{t}}{\left(a + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{b \cdot \sqrt[3]{y}}{t}\right) + 1}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r508274 = x;
        double r508275 = y;
        double r508276 = z;
        double r508277 = r508275 * r508276;
        double r508278 = t;
        double r508279 = r508277 / r508278;
        double r508280 = r508274 + r508279;
        double r508281 = a;
        double r508282 = 1.0;
        double r508283 = r508281 + r508282;
        double r508284 = b;
        double r508285 = r508275 * r508284;
        double r508286 = r508285 / r508278;
        double r508287 = r508283 + r508286;
        double r508288 = r508280 / r508287;
        return r508288;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r508289 = z;
        double r508290 = -4.962196836416018e-210;
        bool r508291 = r508289 <= r508290;
        double r508292 = x;
        double r508293 = y;
        double r508294 = t;
        double r508295 = r508293 / r508294;
        double r508296 = cbrt(r508295);
        double r508297 = r508296 * r508296;
        double r508298 = r508289 * r508296;
        double r508299 = r508297 * r508298;
        double r508300 = r508292 + r508299;
        double r508301 = b;
        double r508302 = r508301 * r508295;
        double r508303 = a;
        double r508304 = r508302 + r508303;
        double r508305 = 1.0;
        double r508306 = r508304 + r508305;
        double r508307 = r508300 / r508306;
        double r508308 = 2.1230875330927117e-209;
        bool r508309 = r508289 <= r508308;
        double r508310 = cbrt(r508293);
        double r508311 = r508310 * r508310;
        double r508312 = r508310 * r508289;
        double r508313 = r508312 / r508294;
        double r508314 = r508311 * r508313;
        double r508315 = r508292 + r508314;
        double r508316 = r508301 * r508310;
        double r508317 = r508316 / r508294;
        double r508318 = r508311 * r508317;
        double r508319 = r508303 + r508318;
        double r508320 = r508319 + r508305;
        double r508321 = r508315 / r508320;
        double r508322 = r508289 / r508294;
        double r508323 = r508293 * r508322;
        double r508324 = r508292 + r508323;
        double r508325 = r508324 / r508306;
        double r508326 = r508309 ? r508321 : r508325;
        double r508327 = r508291 ? r508307 : r508326;
        return r508327;
}

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

Original16.7
Target13.0
Herbie14.7
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -4.962196836416018e-210

    1. Initial program 18.8

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

    2. Simplified15.3

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

    4. Applied add-cube-cbrt15.6

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\frac{y}{t}} \cdot \sqrt[3]{\frac{y}{t}}\right) \cdot \sqrt[3]{\frac{y}{t}}\right)} \cdot z + x}{1 + \left(a + \frac{y}{t} \cdot b\right)}\]

    5. Applied associate-*l*15.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{y}{t}} \cdot \sqrt[3]{\frac{y}{t}}\right) \cdot \left(\sqrt[3]{\frac{y}{t}} \cdot z\right)} + x}{1 + \left(a + \frac{y}{t} \cdot b\right)}\]

    6. Simplified15.6

      \[\leadsto \frac{\left(\sqrt[3]{\frac{y}{t}} \cdot \sqrt[3]{\frac{y}{t}}\right) \cdot \color{blue}{\left(z \cdot \sqrt[3]{\frac{y}{t}}\right)} + x}{1 + \left(a + \frac{y}{t} \cdot b\right)}\]

    if -4.962196836416018e-210 < z < 2.1230875330927117e-209

    1. Initial program 5.9

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

    2. Simplified9.6

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

    4. Applied div-inv9.6

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

    5. Applied associate-*l*6.8

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

    6. Simplified6.8

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

    8. Applied *-un-lft-identity6.8

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

    9. Applied add-cube-cbrt7.0

      \[\leadsto \frac{y \cdot \frac{z}{t} + x}{1 + \left(a + \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{1 \cdot t} \cdot b\right)}\]

    10. Applied times-frac7.0

      \[\leadsto \frac{y \cdot \frac{z}{t} + x}{1 + \left(a + \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{\sqrt[3]{y}}{t}\right)} \cdot b\right)}\]

    11. Applied associate-*l*5.5

      \[\leadsto \frac{y \cdot \frac{z}{t} + x}{1 + \left(a + \color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \left(\frac{\sqrt[3]{y}}{t} \cdot b\right)}\right)}\]

    12. Simplified5.4

      \[\leadsto \frac{y \cdot \frac{z}{t} + x}{1 + \left(a + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \color{blue}{\frac{b \cdot \sqrt[3]{y}}{t}}\right)}\]
    13. Using strategy rm

    14. Applied add-cube-cbrt5.4

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} \cdot \frac{z}{t} + x}{1 + \left(a + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{b \cdot \sqrt[3]{y}}{t}\right)}\]

    15. Applied associate-*l*5.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot \frac{z}{t}\right)} + x}{1 + \left(a + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{b \cdot \sqrt[3]{y}}{t}\right)}\]

    16. Simplified5.0

      \[\leadsto \frac{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \color{blue}{\frac{\sqrt[3]{y} \cdot z}{t}} + x}{1 + \left(a + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{b \cdot \sqrt[3]{y}}{t}\right)}\]

    if 2.1230875330927117e-209 < z

    1. Initial program 18.9

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

    2. Simplified15.2

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

    4. Applied div-inv15.2

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

    5. Applied associate-*l*17.7

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

    6. Simplified17.7

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

  4. Final simplification14.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.96219683641601833436006384613172315374 \cdot 10^{-210}:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{\frac{y}{t}} \cdot \sqrt[3]{\frac{y}{t}}\right) \cdot \left(z \cdot \sqrt[3]{\frac{y}{t}}\right)}{\left(b \cdot \frac{y}{t} + a\right) + 1}\\ \mathbf{elif}\;z \le 2.123087533092711734733727601124115985261 \cdot 10^{-209}:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{\sqrt[3]{y} \cdot z}{t}}{\left(a + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{b \cdot \sqrt[3]{y}}{t}\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(b \cdot \frac{y}{t} + a\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(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) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

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