Average Error: 10.8 → 1.8
Time: 10.2s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.0099085297249333 \cdot 10^{-90} \lor \neg \left(z \le 2.72035343737030278 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.0099085297249333 \cdot 10^{-90} \lor \neg \left(z \le 2.72035343737030278 \cdot 10^{-27}\right):\\
\;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r889561 = x;
        double r889562 = y;
        double r889563 = z;
        double r889564 = r889562 * r889563;
        double r889565 = r889561 - r889564;
        double r889566 = t;
        double r889567 = a;
        double r889568 = r889567 * r889563;
        double r889569 = r889566 - r889568;
        double r889570 = r889565 / r889569;
        return r889570;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r889571 = z;
        double r889572 = -2.0099085297249333e-90;
        bool r889573 = r889571 <= r889572;
        double r889574 = 2.7203534373703028e-27;
        bool r889575 = r889571 <= r889574;
        double r889576 = !r889575;
        bool r889577 = r889573 || r889576;
        double r889578 = x;
        double r889579 = t;
        double r889580 = a;
        double r889581 = r889580 * r889571;
        double r889582 = r889579 - r889581;
        double r889583 = r889578 / r889582;
        double r889584 = y;
        double r889585 = r889579 / r889571;
        double r889586 = r889585 - r889580;
        double r889587 = r889584 / r889586;
        double r889588 = r889583 - r889587;
        double r889589 = 1.0;
        double r889590 = r889584 * r889571;
        double r889591 = r889578 - r889590;
        double r889592 = r889582 / r889591;
        double r889593 = r889589 / r889592;
        double r889594 = r889577 ? r889588 : r889593;
        return r889594;
}

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

Original10.8
Target1.7
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.0099085297249333e-90 or 2.7203534373703028e-27 < z

    1. Initial program 17.9

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

    3. Applied div-sub17.9

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

    4. Simplified11.5

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

    6. Applied clear-num11.6

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

    8. Applied div-sub11.6

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

    9. Simplified2.7

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

    11. Applied un-div-inv2.6

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

    if -2.0099085297249333e-90 < z < 2.7203534373703028e-27

    1. Initial program 0.1

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

    3. Applied clear-num0.6

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.0099085297249333 \cdot 10^{-90} \lor \neg \left(z \le 2.72035343737030278 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))