Average Error: 10.2 → 10.2
Time: 3.9s
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 r639297 = x;
        double r639298 = y;
        double r639299 = z;
        double r639300 = r639298 * r639299;
        double r639301 = r639297 - r639300;
        double r639302 = t;
        double r639303 = a;
        double r639304 = r639303 * r639299;
        double r639305 = r639302 - r639304;
        double r639306 = r639301 / r639305;
        return r639306;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r639307 = z;
        double r639308 = y;
        double r639309 = x;
        double r639310 = -r639309;
        double r639311 = fma(r639307, r639308, r639310);
        double r639312 = a;
        double r639313 = t;
        double r639314 = -r639313;
        double r639315 = fma(r639307, r639312, r639314);
        double r639316 = r639311 / r639315;
        return r639316;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.2
Target1.7
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.2

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

  3. Applied frac-2neg10.2

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

  4. Simplified10.2

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

  5. Simplified10.2

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

  6. Final simplification10.2

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

Reproduce

herbie shell --seed 2020039 +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))))