Average Error: 10.5 → 3.8
Time: 9.1s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.422147047487083834441783199364442582672 \cdot 10^{-154} \lor \neg \left(z \le 2.571049573594902717921673289474660102313 \cdot 10^{-123}\right):\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x}} - \frac{y}{\frac{t}{z} - \frac{a}{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.422147047487083834441783199364442582672 \cdot 10^{-154} \lor \neg \left(z \le 2.571049573594902717921673289474660102313 \cdot 10^{-123}\right):\\
\;\;\;\;\frac{1}{\frac{t - a \cdot z}{x}} - \frac{y}{\frac{t}{z} - \frac{a}{1}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r696233 = x;
        double r696234 = y;
        double r696235 = z;
        double r696236 = r696234 * r696235;
        double r696237 = r696233 - r696236;
        double r696238 = t;
        double r696239 = a;
        double r696240 = r696239 * r696235;
        double r696241 = r696238 - r696240;
        double r696242 = r696237 / r696241;
        return r696242;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r696243 = z;
        double r696244 = -1.4221470474870838e-154;
        bool r696245 = r696243 <= r696244;
        double r696246 = 2.5710495735949027e-123;
        bool r696247 = r696243 <= r696246;
        double r696248 = !r696247;
        bool r696249 = r696245 || r696248;
        double r696250 = 1.0;
        double r696251 = t;
        double r696252 = a;
        double r696253 = r696252 * r696243;
        double r696254 = r696251 - r696253;
        double r696255 = x;
        double r696256 = r696254 / r696255;
        double r696257 = r696250 / r696256;
        double r696258 = y;
        double r696259 = r696251 / r696243;
        double r696260 = r696252 / r696250;
        double r696261 = r696259 - r696260;
        double r696262 = r696258 / r696261;
        double r696263 = r696257 - r696262;
        double r696264 = r696255 / r696254;
        double r696265 = r696258 * r696243;
        double r696266 = r696265 / r696251;
        double r696267 = r696264 - r696266;
        double r696268 = r696249 ? r696263 : r696267;
        return r696268;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.5
Target1.7
Herbie3.8
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.4221470474870838e-154 or 2.5710495735949027e-123 < z

    1. Initial program 14.4

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

    3. Applied div-sub14.4

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

    5. Applied associate-/l*9.3

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

    7. Applied div-sub9.3

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

    8. Simplified2.6

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

    10. Applied clear-num2.8

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

    if -1.4221470474870838e-154 < z < 2.5710495735949027e-123

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Taylor expanded around 0 6.5

      \[\leadsto \frac{x}{t - a \cdot z} - \frac{y \cdot z}{\color{blue}{t}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.422147047487083834441783199364442582672 \cdot 10^{-154} \lor \neg \left(z \le 2.571049573594902717921673289474660102313 \cdot 10^{-123}\right):\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x}} - \frac{y}{\frac{t}{z} - \frac{a}{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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))))