Average Error: 10.3 → 6.5
Time: 11.7s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.06401190083563283 \cdot 10^{38} \lor \neg \left(z \le 2.50952591182416396 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t - a \cdot z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.06401190083563283 \cdot 10^{38} \lor \neg \left(z \le 2.50952591182416396 \cdot 10^{-42}\right):\\
\;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t - a \cdot z}{z}}\\

\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 r773247 = x;
        double r773248 = y;
        double r773249 = z;
        double r773250 = r773248 * r773249;
        double r773251 = r773247 - r773250;
        double r773252 = t;
        double r773253 = a;
        double r773254 = r773253 * r773249;
        double r773255 = r773252 - r773254;
        double r773256 = r773251 / r773255;
        return r773256;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r773257 = z;
        double r773258 = -4.064011900835633e+38;
        bool r773259 = r773257 <= r773258;
        double r773260 = 2.509525911824164e-42;
        bool r773261 = r773257 <= r773260;
        double r773262 = !r773261;
        bool r773263 = r773259 || r773262;
        double r773264 = x;
        double r773265 = t;
        double r773266 = a;
        double r773267 = r773266 * r773257;
        double r773268 = r773265 - r773267;
        double r773269 = r773264 / r773268;
        double r773270 = y;
        double r773271 = 1.0;
        double r773272 = r773268 / r773257;
        double r773273 = r773271 / r773272;
        double r773274 = r773270 * r773273;
        double r773275 = r773269 - r773274;
        double r773276 = r773270 * r773257;
        double r773277 = r773276 / r773268;
        double r773278 = r773269 - r773277;
        double r773279 = r773263 ? r773275 : r773278;
        return r773279;
}

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

Original10.3
Target1.7
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.064011900835633e+38 or 2.509525911824164e-42 < z

    1. Initial program 20.2

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

    3. Applied div-sub20.2

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

    4. Simplified12.7

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

    6. Applied clear-num12.8

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

    if -4.064011900835633e+38 < z < 2.509525911824164e-42

    1. Initial program 0.3

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

    3. Applied div-sub0.3

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

    4. Simplified2.9

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

    6. Applied associate-*r/0.3

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.06401190083563283 \cdot 10^{38} \lor \neg \left(z \le 2.50952591182416396 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t - a \cdot z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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))))