Average Error: 10.6 → 8.9
Time: 13.3s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.9800017626576447 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t - a \cdot z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t - a \cdot z} \cdot \left(x - y \cdot z\right)\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.9800017626576447 \cdot 10^{-92}:\\
\;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t - a \cdot z}{z}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r37053208 = x;
        double r37053209 = y;
        double r37053210 = z;
        double r37053211 = r37053209 * r37053210;
        double r37053212 = r37053208 - r37053211;
        double r37053213 = t;
        double r37053214 = a;
        double r37053215 = r37053214 * r37053210;
        double r37053216 = r37053213 - r37053215;
        double r37053217 = r37053212 / r37053216;
        return r37053217;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r37053218 = z;
        double r37053219 = -1.9800017626576447e-92;
        bool r37053220 = r37053218 <= r37053219;
        double r37053221 = x;
        double r37053222 = t;
        double r37053223 = a;
        double r37053224 = r37053223 * r37053218;
        double r37053225 = r37053222 - r37053224;
        double r37053226 = r37053221 / r37053225;
        double r37053227 = y;
        double r37053228 = r37053225 / r37053218;
        double r37053229 = r37053227 / r37053228;
        double r37053230 = r37053226 - r37053229;
        double r37053231 = 1.0;
        double r37053232 = r37053231 / r37053225;
        double r37053233 = r37053227 * r37053218;
        double r37053234 = r37053221 - r37053233;
        double r37053235 = r37053232 * r37053234;
        double r37053236 = r37053220 ? r37053230 : r37053235;
        return r37053236;
}

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

Original10.6
Target1.6
Herbie8.9
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344.0:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.9800017626576447e-92

    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. Using strategy rm

    5. Applied associate-/l*11.0

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

    if -1.9800017626576447e-92 < z

    1. Initial program 7.7

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

    3. Applied div-sub7.7

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

    5. Applied div-inv7.7

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

    6. Applied div-inv7.8

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

    7. Applied distribute-rgt-out--7.8

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.9800017626576447 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t - a \cdot z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t - a \cdot z} \cdot \left(x - y \cdot z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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))))