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

Average Error: 3.0 → 3.0

Time: 9.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\frac{-x}{\mathsf{fma}\left(t, z, -y\right)}\]

C

double f(double x, double y, double z, double t) {
        double r726901 = x;
        double r726902 = y;
        double r726903 = z;
        double r726904 = t;
        double r726905 = r726903 * r726904;
        double r726906 = r726902 - r726905;
        double r726907 = r726901 / r726906;
        return r726907;
}

↓

double f(double x, double y, double z, double t) {
        double r726908 = x;
        double r726909 = -r726908;
        double r726910 = t;
        double r726911 = z;
        double r726912 = y;
        double r726913 = -r726912;
        double r726914 = fma(r726910, r726911, r726913);
        double r726915 = r726909 / r726914;
        return r726915;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	3.0
Target	1.7
Herbie	3.0

\[\begin{array}{l} \mathbf{if}\;x \lt -1.618195973607049 \cdot 10^{50}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{elif}\;x \lt 2.13783064348764444 \cdot 10^{131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \end{array}\]

Derivation

1. Initial program 3.0

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

3. Applied frac-2neg 3.0

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

4. Simplified 3.0

   \[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(t, z, -y\right)}}\]

5. Final simplification 3.0

   \[\leadsto \frac{-x}{\mathsf{fma}\left(t, z, -y\right)}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1 (- (/ y x) (* (/ z x) t)))))

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