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

Average Error: 2.7 → 2.7

Time: 13.8s

Precision: 64

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

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r764593 = x;
        double r764594 = y;
        double r764595 = z;
        double r764596 = t;
        double r764597 = r764595 * r764596;
        double r764598 = r764594 - r764597;
        double r764599 = r764593 / r764598;
        return r764599;
}

↓

double f(double x, double y, double z, double t) {
        double r764600 = x;
        double r764601 = -r764600;
        double r764602 = z;
        double r764603 = t;
        double r764604 = y;
        double r764605 = -r764604;
        double r764606 = fma(r764602, r764603, r764605);
        double r764607 = r764601 / r764606;
        return r764607;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.7
Target	1.7
Herbie	2.7

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

Derivation

1. Initial program 2.7

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

3. Applied frac-2neg 2.7

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

4. Simplified 2.7

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

5. Final simplification 2.7

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

Reproduce

herbie shell --seed 2020043 +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))))