Average Error: 10.9 → 1.9
Time: 21.8s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.873159407921233748848095007688082806631 \cdot 10^{59} \lor \neg \left(z \le 1.295167332391820426867656083484373540485 \cdot 10^{-63}\right):\\ \;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.873159407921233748848095007688082806631 \cdot 10^{59} \lor \neg \left(z \le 1.295167332391820426867656083484373540485 \cdot 10^{-63}\right):\\
\;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r411077 = x;
        double r411078 = y;
        double r411079 = z;
        double r411080 = r411078 * r411079;
        double r411081 = r411077 - r411080;
        double r411082 = t;
        double r411083 = a;
        double r411084 = r411083 * r411079;
        double r411085 = r411082 - r411084;
        double r411086 = r411081 / r411085;
        return r411086;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r411087 = z;
        double r411088 = -1.8731594079212337e+59;
        bool r411089 = r411087 <= r411088;
        double r411090 = 1.2951673323918204e-63;
        bool r411091 = r411087 <= r411090;
        double r411092 = !r411091;
        bool r411093 = r411089 || r411092;
        double r411094 = x;
        double r411095 = 1.0;
        double r411096 = t;
        double r411097 = a;
        double r411098 = r411097 * r411087;
        double r411099 = r411096 - r411098;
        double r411100 = r411095 / r411099;
        double r411101 = r411094 * r411100;
        double r411102 = y;
        double r411103 = r411096 / r411087;
        double r411104 = r411103 - r411097;
        double r411105 = r411102 / r411104;
        double r411106 = r411101 - r411105;
        double r411107 = r411094 / r411099;
        double r411108 = r411102 * r411087;
        double r411109 = r411108 / r411099;
        double r411110 = r411107 - r411109;
        double r411111 = r411093 ? r411106 : r411110;
        return r411111;
}

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

Original10.9
Target1.8
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.8731594079212337e+59 or 1.2951673323918204e-63 < z

    1. Initial program 20.6

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

    3. Applied div-sub20.6

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

    4. Simplified13.0

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

    6. Applied pow113.0

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

    7. Applied pow113.0

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

    8. Applied pow-prod-down13.0

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

    9. Simplified3.0

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

    11. Applied div-inv3.0

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

    if -1.8731594079212337e+59 < z < 1.2951673323918204e-63

    1. Initial program 0.6

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

    3. Applied div-sub0.6

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

    4. Simplified2.7

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

    6. Applied associate-*r/0.6

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.873159407921233748848095007688082806631 \cdot 10^{59} \lor \neg \left(z \le 1.295167332391820426867656083484373540485 \cdot 10^{-63}\right):\\ \;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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 3.51395223729782958e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))