Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A

Average Error: 10.5 → 10.6

Time: 19.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r36358654 = x;
        double r36358655 = y;
        double r36358656 = z;
        double r36358657 = r36358655 * r36358656;
        double r36358658 = r36358654 - r36358657;
        double r36358659 = t;
        double r36358660 = a;
        double r36358661 = r36358660 * r36358656;
        double r36358662 = r36358659 - r36358661;
        double r36358663 = r36358658 / r36358662;
        return r36358663;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r36358664 = x;
        double r36358665 = z;
        double r36358666 = y;
        double r36358667 = r36358665 * r36358666;
        double r36358668 = r36358664 - r36358667;
        double r36358669 = 1.0;
        double r36358670 = t;
        double r36358671 = a;
        double r36358672 = r36358671 * r36358665;
        double r36358673 = r36358670 - r36358672;
        double r36358674 = r36358669 / r36358673;
        double r36358675 = r36358668 * r36358674;
        return r36358675;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.5
Target	1.8
Herbie	10.6

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

Derivation

1. Initial program 10.5

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

3. Applied div-inv 10.6

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

4. Final simplification 10.6

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(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)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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