Average Error: 10.6 → 10.6
Time: 4.3s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{\mathsf{fma}\left(z, y, -x\right)}{\mathsf{fma}\left(z, a, -t\right)}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{\mathsf{fma}\left(z, y, -x\right)}{\mathsf{fma}\left(z, a, -t\right)}
double f(double x, double y, double z, double t, double a) {
        double r681449 = x;
        double r681450 = y;
        double r681451 = z;
        double r681452 = r681450 * r681451;
        double r681453 = r681449 - r681452;
        double r681454 = t;
        double r681455 = a;
        double r681456 = r681455 * r681451;
        double r681457 = r681454 - r681456;
        double r681458 = r681453 / r681457;
        return r681458;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r681459 = z;
        double r681460 = y;
        double r681461 = x;
        double r681462 = -r681461;
        double r681463 = fma(r681459, r681460, r681462);
        double r681464 = a;
        double r681465 = t;
        double r681466 = -r681465;
        double r681467 = fma(r681459, r681464, r681466);
        double r681468 = r681463 / r681467;
        return r681468;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.6
Target1.7
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.6

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

  3. Applied frac-2neg10.6

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

  4. Simplified10.6

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

  5. Simplified10.6

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

  6. Final simplification10.6

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

Reproduce

herbie shell --seed 2020089 +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))))