Average Error: 10.9 → 11.0
Time: 3.3s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\mathsf{fma}\left(z, y, -x\right) \cdot \frac{1}{\mathsf{fma}\left(z, a, -t\right)}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\mathsf{fma}\left(z, y, -x\right) \cdot \frac{1}{\mathsf{fma}\left(z, a, -t\right)}
double f(double x, double y, double z, double t, double a) {
        double r671346 = x;
        double r671347 = y;
        double r671348 = z;
        double r671349 = r671347 * r671348;
        double r671350 = r671346 - r671349;
        double r671351 = t;
        double r671352 = a;
        double r671353 = r671352 * r671348;
        double r671354 = r671351 - r671353;
        double r671355 = r671350 / r671354;
        return r671355;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r671356 = z;
        double r671357 = y;
        double r671358 = x;
        double r671359 = -r671358;
        double r671360 = fma(r671356, r671357, r671359);
        double r671361 = 1.0;
        double r671362 = a;
        double r671363 = t;
        double r671364 = -r671363;
        double r671365 = fma(r671356, r671362, r671364);
        double r671366 = r671361 / r671365;
        double r671367 = r671360 * r671366;
        return r671367;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.9
Target1.8
Herbie11.0
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.9

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

  3. Applied frac-2neg10.9

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

  4. Simplified10.9

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

  5. Simplified10.9

    \[\leadsto \frac{\mathsf{fma}\left(z, y, -x\right)}{\color{blue}{\mathsf{fma}\left(z, a, -t\right)}}\]
  6. Using strategy rm

  7. Applied div-inv11.0

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

  8. Final simplification11.0

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

Reproduce

herbie shell --seed 2020036 +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))))