Average Error: 10.1 → 10.2
Time: 10.1s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\left(x - y \cdot z\right) \cdot \frac{1}{\mathsf{fma}\left(0, z, \mathsf{fma}\left(a, -z, t\right)\right)}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\left(x - y \cdot z\right) \cdot \frac{1}{\mathsf{fma}\left(0, z, \mathsf{fma}\left(a, -z, t\right)\right)}
double f(double x, double y, double z, double t, double a) {
        double r539691 = x;
        double r539692 = y;
        double r539693 = z;
        double r539694 = r539692 * r539693;
        double r539695 = r539691 - r539694;
        double r539696 = t;
        double r539697 = a;
        double r539698 = r539697 * r539693;
        double r539699 = r539696 - r539698;
        double r539700 = r539695 / r539699;
        return r539700;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r539701 = x;
        double r539702 = y;
        double r539703 = z;
        double r539704 = r539702 * r539703;
        double r539705 = r539701 - r539704;
        double r539706 = 1.0;
        double r539707 = 0.0;
        double r539708 = a;
        double r539709 = -r539703;
        double r539710 = t;
        double r539711 = fma(r539708, r539709, r539710);
        double r539712 = fma(r539707, r539703, r539711);
        double r539713 = r539706 / r539712;
        double r539714 = r539705 * r539713;
        return r539714;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.1
Target1.6
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.1

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

  3. Applied add-sqr-sqrt37.5

    \[\leadsto \frac{x - y \cdot z}{\color{blue}{\sqrt{t} \cdot \sqrt{t}} - a \cdot z}\]

  4. Applied prod-diff38.5

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

  5. Simplified12.1

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

  6. Simplified10.1

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

  8. Applied div-inv10.2

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

  9. Simplified10.2

    \[\leadsto \left(x - y \cdot z\right) \cdot \color{blue}{\frac{1}{\mathsf{fma}\left(0, z, \mathsf{fma}\left(a, -z, t\right)\right)}}\]

  10. Final simplification10.2

    \[\leadsto \left(x - y \cdot z\right) \cdot \frac{1}{\mathsf{fma}\left(0, z, \mathsf{fma}\left(a, -z, t\right)\right)}\]

Reproduce

herbie shell --seed 2019212 +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 3.51395223729782958e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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