Average Error: 10.6 → 7.2
Time: 10.1s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{x}{t - a \cdot z} - \left(y \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t - a \cdot z}}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{x}{t - a \cdot z} - \left(y \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t - a \cdot z}}
double f(double x, double y, double z, double t, double a) {
        double r449297 = x;
        double r449298 = y;
        double r449299 = z;
        double r449300 = r449298 * r449299;
        double r449301 = r449297 - r449300;
        double r449302 = t;
        double r449303 = a;
        double r449304 = r449303 * r449299;
        double r449305 = r449302 - r449304;
        double r449306 = r449301 / r449305;
        return r449306;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r449307 = x;
        double r449308 = t;
        double r449309 = a;
        double r449310 = z;
        double r449311 = r449309 * r449310;
        double r449312 = r449308 - r449311;
        double r449313 = r449307 / r449312;
        double r449314 = y;
        double r449315 = cbrt(r449310);
        double r449316 = r449315 * r449315;
        double r449317 = cbrt(r449312);
        double r449318 = r449317 * r449317;
        double r449319 = r449316 / r449318;
        double r449320 = r449314 * r449319;
        double r449321 = r449315 / r449317;
        double r449322 = r449320 * r449321;
        double r449323 = r449313 - r449322;
        return r449323;
}

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
Herbie7.2
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.6

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

  3. Applied div-sub10.6

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

  5. Applied *-un-lft-identity10.6

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

  6. Applied times-frac7.9

    \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{\frac{y}{1} \cdot \frac{z}{t - a \cdot z}}\]

  7. Simplified7.9

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

  9. Applied add-cube-cbrt8.2

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

  10. Applied add-cube-cbrt8.3

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

  11. Applied times-frac8.3

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

  12. Applied associate-*r*7.2

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

  13. Final simplification7.2

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

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))))