Average Error: 10.0 → 1.7
Time: 12.0s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -9.2159596611983673 \cdot 10^{-15} \lor \neg \left(z \le 3.82302836807757522 \cdot 10^{-132}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a, -z, t\right)}{x - z \cdot y}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -9.2159596611983673 \cdot 10^{-15} \lor \neg \left(z \le 3.82302836807757522 \cdot 10^{-132}\right):\\
\;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a, -z, t\right)}{x - z \cdot y}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r786884 = x;
        double r786885 = y;
        double r786886 = z;
        double r786887 = r786885 * r786886;
        double r786888 = r786884 - r786887;
        double r786889 = t;
        double r786890 = a;
        double r786891 = r786890 * r786886;
        double r786892 = r786889 - r786891;
        double r786893 = r786888 / r786892;
        return r786893;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r786894 = z;
        double r786895 = -9.215959661198367e-15;
        bool r786896 = r786894 <= r786895;
        double r786897 = 3.823028368077575e-132;
        bool r786898 = r786894 <= r786897;
        double r786899 = !r786898;
        bool r786900 = r786896 || r786899;
        double r786901 = x;
        double r786902 = t;
        double r786903 = a;
        double r786904 = r786903 * r786894;
        double r786905 = r786902 - r786904;
        double r786906 = r786901 / r786905;
        double r786907 = y;
        double r786908 = r786902 / r786894;
        double r786909 = r786908 - r786903;
        double r786910 = r786907 / r786909;
        double r786911 = r786906 - r786910;
        double r786912 = 1.0;
        double r786913 = -r786894;
        double r786914 = fma(r786903, r786913, r786902);
        double r786915 = r786894 * r786907;
        double r786916 = r786901 - r786915;
        double r786917 = r786914 / r786916;
        double r786918 = r786912 / r786917;
        double r786919 = r786900 ? r786911 : r786918;
        return r786919;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.0
Target1.5
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -9.215959661198367e-15 or 3.823028368077575e-132 < z

    1. Initial program 16.4

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

    3. Applied div-sub16.4

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

    4. Simplified10.5

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

    6. Applied clear-num10.6

      \[\leadsto \frac{x}{t - a \cdot z} - y \cdot \color{blue}{\frac{1}{\frac{t - z \cdot a}{z}}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity10.6

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

    9. Applied associate-*l*10.6

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

    10. Simplified2.5

      \[\leadsto \frac{x}{t - a \cdot z} - 1 \cdot \color{blue}{\frac{y}{\frac{t}{z} - a}}\]

    if -9.215959661198367e-15 < z < 3.823028368077575e-132

    1. Initial program 0.1

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

    3. Applied clear-num0.5

      \[\leadsto \color{blue}{\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}}\]

    4. Simplified0.5

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(a, -z, t\right)}{x - z \cdot y}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.2159596611983673 \cdot 10^{-15} \lor \neg \left(z \le 3.82302836807757522 \cdot 10^{-132}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a, -z, t\right)}{x - z \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))))