Average Error: 10.6 → 10.7
Time: 5.1s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{1}{t - a \cdot z} \cdot \left(x - y \cdot z\right)\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{1}{t - a \cdot z} \cdot \left(x - y \cdot z\right)
double f(double x, double y, double z, double t, double a) {
        double r1005349 = x;
        double r1005350 = y;
        double r1005351 = z;
        double r1005352 = r1005350 * r1005351;
        double r1005353 = r1005349 - r1005352;
        double r1005354 = t;
        double r1005355 = a;
        double r1005356 = r1005355 * r1005351;
        double r1005357 = r1005354 - r1005356;
        double r1005358 = r1005353 / r1005357;
        return r1005358;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r1005359 = 1.0;
        double r1005360 = t;
        double r1005361 = a;
        double r1005362 = z;
        double r1005363 = r1005361 * r1005362;
        double r1005364 = r1005360 - r1005363;
        double r1005365 = r1005359 / r1005364;
        double r1005366 = x;
        double r1005367 = y;
        double r1005368 = r1005367 * r1005362;
        double r1005369 = r1005366 - r1005368;
        double r1005370 = r1005365 * r1005369;
        return r1005370;
}

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
Target1.8
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.6

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

  3. Applied clear-num10.9

    \[\leadsto \color{blue}{\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}}\]
  4. Using strategy rm

  5. Applied div-inv11.0

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

  6. Applied associate-/r*10.8

    \[\leadsto \color{blue}{\frac{\frac{1}{t - a \cdot z}}{\frac{1}{x - y \cdot z}}}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity10.8

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

  9. Applied add-sqr-sqrt10.8

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

  10. Applied times-frac10.8

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

  11. Applied *-un-lft-identity10.8

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

  12. Applied add-sqr-sqrt10.8

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

  13. Applied times-frac10.8

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

  14. Applied times-frac10.8

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

  15. Simplified10.8

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

  16. Simplified10.7

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

  17. Final simplification10.7

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(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 3.5139522372978296e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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