Average Error: 10.6 → 1.9
Time: 20.8s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -469735603.148817598819732666015625 \lor \neg \left(z \le 3.061405469162073408418218211607294509253 \cdot 10^{53}\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{x}{t - a \cdot z}} \cdot \sqrt[3]{\frac{x}{t - a \cdot z}}\right) \cdot \sqrt[3]{\frac{x}{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -469735603.148817598819732666015625 \lor \neg \left(z \le 3.061405469162073408418218211607294509253 \cdot 10^{53}\right):\\
\;\;\;\;\left(\sqrt[3]{\frac{x}{t - a \cdot z}} \cdot \sqrt[3]{\frac{x}{t - a \cdot z}}\right) \cdot \sqrt[3]{\frac{x}{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r497294 = x;
        double r497295 = y;
        double r497296 = z;
        double r497297 = r497295 * r497296;
        double r497298 = r497294 - r497297;
        double r497299 = t;
        double r497300 = a;
        double r497301 = r497300 * r497296;
        double r497302 = r497299 - r497301;
        double r497303 = r497298 / r497302;
        return r497303;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r497304 = z;
        double r497305 = -469735603.1488176;
        bool r497306 = r497304 <= r497305;
        double r497307 = 3.0614054691620734e+53;
        bool r497308 = r497304 <= r497307;
        double r497309 = !r497308;
        bool r497310 = r497306 || r497309;
        double r497311 = x;
        double r497312 = t;
        double r497313 = a;
        double r497314 = r497313 * r497304;
        double r497315 = r497312 - r497314;
        double r497316 = r497311 / r497315;
        double r497317 = cbrt(r497316);
        double r497318 = r497317 * r497317;
        double r497319 = r497318 * r497317;
        double r497320 = y;
        double r497321 = r497312 / r497304;
        double r497322 = r497321 - r497313;
        double r497323 = r497320 / r497322;
        double r497324 = r497319 - r497323;
        double r497325 = 1.0;
        double r497326 = r497320 * r497304;
        double r497327 = r497311 - r497326;
        double r497328 = r497315 / r497327;
        double r497329 = r497325 / r497328;
        double r497330 = r497310 ? r497324 : r497329;
        return r497330;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.6
Target1.6
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -469735603.1488176 or 3.0614054691620734e+53 < z

    1. Initial program 22.7

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
    2. Using strategy rm

    3. Applied div-sub22.7

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

    4. Simplified14.1

      \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{y \cdot \frac{z}{t - a \cdot z}}\]
    5. Using strategy rm

    6. Applied pow114.1

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

    7. Applied pow114.1

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

    8. Applied pow-prod-down14.1

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

    9. Simplified3.0

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

    11. Applied add-cube-cbrt3.2

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

    if -469735603.1488176 < z < 3.0614054691620734e+53

    1. Initial program 0.4

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
    2. Using strategy rm

    3. Applied clear-num0.9

      \[\leadsto \color{blue}{\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -469735603.148817598819732666015625 \lor \neg \left(z \le 3.061405469162073408418218211607294509253 \cdot 10^{53}\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{x}{t - a \cdot z}} \cdot \sqrt[3]{\frac{x}{t - a \cdot z}}\right) \cdot \sqrt[3]{\frac{x}{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]

Reproduce

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

  :herbie-target
  (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.51395223729782958e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))