Average Error: 10.2 → 2.9
Time: 20.4s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{x}{t - a \cdot z} - \frac{y}{\left(t \cdot \frac{1}{z} - a\right) + \mathsf{fma}\left(a, -1, a\right)}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{x}{t - a \cdot z} - \frac{y}{\left(t \cdot \frac{1}{z} - a\right) + \mathsf{fma}\left(a, -1, a\right)}
double f(double x, double y, double z, double t, double a) {
        double r482689 = x;
        double r482690 = y;
        double r482691 = z;
        double r482692 = r482690 * r482691;
        double r482693 = r482689 - r482692;
        double r482694 = t;
        double r482695 = a;
        double r482696 = r482695 * r482691;
        double r482697 = r482694 - r482696;
        double r482698 = r482693 / r482697;
        return r482698;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r482699 = x;
        double r482700 = t;
        double r482701 = a;
        double r482702 = z;
        double r482703 = r482701 * r482702;
        double r482704 = r482700 - r482703;
        double r482705 = r482699 / r482704;
        double r482706 = y;
        double r482707 = 1.0;
        double r482708 = r482707 / r482702;
        double r482709 = r482700 * r482708;
        double r482710 = r482709 - r482701;
        double r482711 = -1.0;
        double r482712 = fma(r482701, r482711, r482701);
        double r482713 = r482710 + r482712;
        double r482714 = r482706 / r482713;
        double r482715 = r482705 - r482714;
        return r482715;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.2
Target1.8
Herbie2.9
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.2

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

  3. Applied div-sub10.2

    \[\leadsto \color{blue}{\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}}\]
  4. Using strategy rm

  5. Applied associate-/l*7.6

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

  6. Taylor expanded around 0 2.8

    \[\leadsto \frac{x}{t - a \cdot z} - \frac{y}{\color{blue}{\frac{t}{z} - a}}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt3.2

    \[\leadsto \frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - \color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

  9. Applied div-inv3.2

    \[\leadsto \frac{x}{t - a \cdot z} - \frac{y}{\color{blue}{t \cdot \frac{1}{z}} - \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

  10. Applied prod-diff3.2

    \[\leadsto \frac{x}{t - a \cdot z} - \frac{y}{\color{blue}{\mathsf{fma}\left(t, \frac{1}{z}, -\sqrt[3]{a} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{a}, \sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right)}}\]

  11. Simplified2.9

    \[\leadsto \frac{x}{t - a \cdot z} - \frac{y}{\color{blue}{\left(t \cdot \frac{1}{z} - a\right)} + \mathsf{fma}\left(-\sqrt[3]{a}, \sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right)}\]

  12. Simplified2.9

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

  13. Final simplification2.9

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

Reproduce

herbie shell --seed 2019208 +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))))