Average Error: 10.4 → 2.8
Time: 24.7s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{-x}{\mathsf{fma}\left(z, a, -t\right)} - \frac{y}{\frac{t}{z} - a}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{-x}{\mathsf{fma}\left(z, a, -t\right)} - \frac{y}{\frac{t}{z} - a}
double f(double x, double y, double z, double t, double a) {
        double r464211 = x;
        double r464212 = y;
        double r464213 = z;
        double r464214 = r464212 * r464213;
        double r464215 = r464211 - r464214;
        double r464216 = t;
        double r464217 = a;
        double r464218 = r464217 * r464213;
        double r464219 = r464216 - r464218;
        double r464220 = r464215 / r464219;
        return r464220;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r464221 = x;
        double r464222 = -r464221;
        double r464223 = z;
        double r464224 = a;
        double r464225 = t;
        double r464226 = -r464225;
        double r464227 = fma(r464223, r464224, r464226);
        double r464228 = r464222 / r464227;
        double r464229 = y;
        double r464230 = r464225 / r464223;
        double r464231 = r464230 - r464224;
        double r464232 = r464229 / r464231;
        double r464233 = r464228 - r464232;
        return r464233;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.4
Target1.8
Herbie2.8
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.4

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

  3. Applied div-sub10.4

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

  4. Simplified7.8

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

  6. Applied frac-2neg7.8

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

  7. Simplified7.8

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

  9. Applied clear-num7.9

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

  11. Applied *-un-lft-identity7.9

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

  12. Applied associate-*l*7.9

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

  13. Simplified2.8

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

  14. Final simplification2.8

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

Reproduce

herbie shell --seed 2019326 +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))))