Average Error: 10.3 → 8.8
Time: 16.2s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{x}{t - a \cdot z} - \frac{y}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}} \cdot \frac{z}{\sqrt[3]{\left(\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}\right) \cdot \sqrt[3]{t - a \cdot z}}}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{x}{t - a \cdot z} - \frac{y}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}} \cdot \frac{z}{\sqrt[3]{\left(\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}\right) \cdot \sqrt[3]{t - a \cdot z}}}
double f(double x, double y, double z, double t, double a) {
        double r38618488 = x;
        double r38618489 = y;
        double r38618490 = z;
        double r38618491 = r38618489 * r38618490;
        double r38618492 = r38618488 - r38618491;
        double r38618493 = t;
        double r38618494 = a;
        double r38618495 = r38618494 * r38618490;
        double r38618496 = r38618493 - r38618495;
        double r38618497 = r38618492 / r38618496;
        return r38618497;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r38618498 = x;
        double r38618499 = t;
        double r38618500 = a;
        double r38618501 = z;
        double r38618502 = r38618500 * r38618501;
        double r38618503 = r38618499 - r38618502;
        double r38618504 = r38618498 / r38618503;
        double r38618505 = y;
        double r38618506 = cbrt(r38618503);
        double r38618507 = r38618506 * r38618506;
        double r38618508 = r38618505 / r38618507;
        double r38618509 = r38618507 * r38618506;
        double r38618510 = cbrt(r38618509);
        double r38618511 = r38618501 / r38618510;
        double r38618512 = r38618508 * r38618511;
        double r38618513 = r38618504 - r38618512;
        return r38618513;
}

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

Original10.3
Target1.8
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344.0:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.3

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

  3. Applied div-sub10.3

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

  5. Applied add-cube-cbrt10.6

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

  6. Applied times-frac8.8

    \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{\frac{y}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}} \cdot \frac{z}{\sqrt[3]{t - a \cdot z}}}\]
  7. Using strategy rm

  8. Applied add-cbrt-cube8.8

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

  9. Final simplification8.8

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

Reproduce

herbie shell --seed 2019162 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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