Average Error: 10.9 → 9.5
Time: 57.4s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.190032178478921 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t - a \cdot z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{\sqrt[3]{x}}{\sqrt[3]{t - a \cdot z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{t - a \cdot z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{t - a \cdot z}}\right), \frac{-1}{t - a \cdot z} \cdot \left(y \cdot z\right)\right) + \mathsf{fma}\left(\frac{-1}{t - a \cdot z}, y \cdot z, \frac{1}{t - a \cdot z} \cdot \left(y \cdot z\right)\right)\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -6.190032178478921 \cdot 10^{-100}:\\
\;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t - a \cdot z}{z}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{\sqrt[3]{x}}{\sqrt[3]{t - a \cdot z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{t - a \cdot z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{t - a \cdot z}}\right), \frac{-1}{t - a \cdot z} \cdot \left(y \cdot z\right)\right) + \mathsf{fma}\left(\frac{-1}{t - a \cdot z}, y \cdot z, \frac{1}{t - a \cdot z} \cdot \left(y \cdot z\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r25439629 = x;
        double r25439630 = y;
        double r25439631 = z;
        double r25439632 = r25439630 * r25439631;
        double r25439633 = r25439629 - r25439632;
        double r25439634 = t;
        double r25439635 = a;
        double r25439636 = r25439635 * r25439631;
        double r25439637 = r25439634 - r25439636;
        double r25439638 = r25439633 / r25439637;
        return r25439638;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r25439639 = z;
        double r25439640 = -6.190032178478921e-100;
        bool r25439641 = r25439639 <= r25439640;
        double r25439642 = x;
        double r25439643 = t;
        double r25439644 = a;
        double r25439645 = r25439644 * r25439639;
        double r25439646 = r25439643 - r25439645;
        double r25439647 = r25439642 / r25439646;
        double r25439648 = y;
        double r25439649 = r25439646 / r25439639;
        double r25439650 = r25439648 / r25439649;
        double r25439651 = r25439647 - r25439650;
        double r25439652 = 1.0;
        double r25439653 = cbrt(r25439642);
        double r25439654 = cbrt(r25439646);
        double r25439655 = r25439653 / r25439654;
        double r25439656 = r25439655 * r25439655;
        double r25439657 = r25439655 * r25439656;
        double r25439658 = -1.0;
        double r25439659 = r25439658 / r25439646;
        double r25439660 = r25439648 * r25439639;
        double r25439661 = r25439659 * r25439660;
        double r25439662 = fma(r25439652, r25439657, r25439661);
        double r25439663 = r25439652 / r25439646;
        double r25439664 = r25439663 * r25439660;
        double r25439665 = fma(r25439659, r25439660, r25439664);
        double r25439666 = r25439662 + r25439665;
        double r25439667 = r25439641 ? r25439651 : r25439666;
        return r25439667;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.9
Target1.6
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344.0:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1.0}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -6.190032178478921e-100

    1. Initial program 16.5

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

    3. Applied div-sub16.5

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

    5. Applied associate-/l*10.8

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

    if -6.190032178478921e-100 < z

    1. Initial program 8.0

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

    3. Applied div-sub8.0

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

    5. Applied div-inv8.0

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

    6. Applied *-un-lft-identity8.0

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

    7. Applied prod-diff8.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{x}{t - a \cdot z}, -\frac{1}{t - a \cdot z} \cdot \left(y \cdot z\right)\right) + \mathsf{fma}\left(-\frac{1}{t - a \cdot z}, y \cdot z, \frac{1}{t - a \cdot z} \cdot \left(y \cdot z\right)\right)}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt8.6

      \[\leadsto \mathsf{fma}\left(1, \frac{x}{\color{blue}{\left(\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}\right) \cdot \sqrt[3]{t - a \cdot z}}}, -\frac{1}{t - a \cdot z} \cdot \left(y \cdot z\right)\right) + \mathsf{fma}\left(-\frac{1}{t - a \cdot z}, y \cdot z, \frac{1}{t - a \cdot z} \cdot \left(y \cdot z\right)\right)\]

    10. Applied add-cube-cbrt8.8

      \[\leadsto \mathsf{fma}\left(1, \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}\right) \cdot \sqrt[3]{t - a \cdot z}}, -\frac{1}{t - a \cdot z} \cdot \left(y \cdot z\right)\right) + \mathsf{fma}\left(-\frac{1}{t - a \cdot z}, y \cdot z, \frac{1}{t - a \cdot z} \cdot \left(y \cdot z\right)\right)\]

    11. Applied times-frac8.8

      \[\leadsto \mathsf{fma}\left(1, \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{t - a \cdot z}}}, -\frac{1}{t - a \cdot z} \cdot \left(y \cdot z\right)\right) + \mathsf{fma}\left(-\frac{1}{t - a \cdot z}, y \cdot z, \frac{1}{t - a \cdot z} \cdot \left(y \cdot z\right)\right)\]

    12. Simplified8.8

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.190032178478921 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t - a \cdot z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{\sqrt[3]{x}}{\sqrt[3]{t - a \cdot z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{t - a \cdot z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{t - a \cdot z}}\right), \frac{-1}{t - a \cdot z} \cdot \left(y \cdot z\right)\right) + \mathsf{fma}\left(\frac{-1}{t - a \cdot z}, y \cdot z, \frac{1}{t - a \cdot z} \cdot \left(y \cdot z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(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)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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