Average Error: 10.3 → 10.3
Time: 19.5s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{-\left(x - z \cdot y\right)}{\mathsf{fma}\left(a, z, -t\right)}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{-\left(x - z \cdot y\right)}{\mathsf{fma}\left(a, z, -t\right)}
double f(double x, double y, double z, double t, double a) {
        double r30993630 = x;
        double r30993631 = y;
        double r30993632 = z;
        double r30993633 = r30993631 * r30993632;
        double r30993634 = r30993630 - r30993633;
        double r30993635 = t;
        double r30993636 = a;
        double r30993637 = r30993636 * r30993632;
        double r30993638 = r30993635 - r30993637;
        double r30993639 = r30993634 / r30993638;
        return r30993639;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r30993640 = x;
        double r30993641 = z;
        double r30993642 = y;
        double r30993643 = r30993641 * r30993642;
        double r30993644 = r30993640 - r30993643;
        double r30993645 = -r30993644;
        double r30993646 = a;
        double r30993647 = t;
        double r30993648 = -r30993647;
        double r30993649 = fma(r30993646, r30993641, r30993648);
        double r30993650 = r30993645 / r30993649;
        return r30993650;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.3
Target1.8
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344.0:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.3

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

  3. Applied frac-2neg10.3

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

  4. Simplified10.3

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

  5. Final simplification10.3

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ 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))))