Average Error: 10.0 → 1.4
Time: 11.0s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.10185718647149596 \cdot 10^{-19} \lor \neg \left(z \le 2.3170079174536806 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.10185718647149596 \cdot 10^{-19} \lor \neg \left(z \le 2.3170079174536806 \cdot 10^{-12}\right):\\
\;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\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 r662015 = x;
        double r662016 = y;
        double r662017 = z;
        double r662018 = r662016 * r662017;
        double r662019 = r662015 - r662018;
        double r662020 = t;
        double r662021 = a;
        double r662022 = r662021 * r662017;
        double r662023 = r662020 - r662022;
        double r662024 = r662019 / r662023;
        return r662024;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r662025 = z;
        double r662026 = -3.101857186471496e-19;
        bool r662027 = r662025 <= r662026;
        double r662028 = 2.3170079174536806e-12;
        bool r662029 = r662025 <= r662028;
        double r662030 = !r662029;
        bool r662031 = r662027 || r662030;
        double r662032 = x;
        double r662033 = t;
        double r662034 = a;
        double r662035 = r662034 * r662025;
        double r662036 = r662033 - r662035;
        double r662037 = r662032 / r662036;
        double r662038 = y;
        double r662039 = 1.0;
        double r662040 = r662033 / r662025;
        double r662041 = r662040 - r662034;
        double r662042 = r662039 / r662041;
        double r662043 = r662038 * r662042;
        double r662044 = r662037 - r662043;
        double r662045 = r662038 * r662025;
        double r662046 = r662045 / r662036;
        double r662047 = r662037 - r662046;
        double r662048 = r662031 ? r662044 : r662047;
        return r662048;
}

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

Original10.0
Target1.5
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.101857186471496e-19 or 2.3170079174536806e-12 < z

    1. Initial program 19.4

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

    3. Applied div-sub19.4

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

    4. Simplified12.0

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

    6. Applied clear-num12.0

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

    8. Applied div-sub12.0

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

    9. Simplified2.6

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

    if -3.101857186471496e-19 < z < 2.3170079174536806e-12

    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. Simplified3.1

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

    6. Applied associate-*r/0.1

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.10185718647149596 \cdot 10^{-19} \lor \neg \left(z \le 2.3170079174536806 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))))