Average Error: 10.4 → 2.0
Time: 17.8s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \le -3.653524365648226522727694876162942276508 \cdot 10^{306} \lor \neg \left(\frac{x - y \cdot z}{t - a \cdot z} \le -7.257659457820581593157072962696615977206 \cdot 10^{-315} \lor \neg \left(\frac{x - y \cdot z}{t - a \cdot z} \le 0.0\right) \land \frac{x - y \cdot z}{t - a \cdot z} \le 5.987964510668616566856227864091206028983 \cdot 10^{295}\right):\\ \;\;\;\;\sqrt[3]{{\left(\frac{x}{t - a \cdot z}\right)}^{3}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \le -3.653524365648226522727694876162942276508 \cdot 10^{306} \lor \neg \left(\frac{x - y \cdot z}{t - a \cdot z} \le -7.257659457820581593157072962696615977206 \cdot 10^{-315} \lor \neg \left(\frac{x - y \cdot z}{t - a \cdot z} \le 0.0\right) \land \frac{x - y \cdot z}{t - a \cdot z} \le 5.987964510668616566856227864091206028983 \cdot 10^{295}\right):\\
\;\;\;\;\sqrt[3]{{\left(\frac{x}{t - a \cdot z}\right)}^{3}} - \frac{y}{\frac{t}{z} - a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r495683 = x;
        double r495684 = y;
        double r495685 = z;
        double r495686 = r495684 * r495685;
        double r495687 = r495683 - r495686;
        double r495688 = t;
        double r495689 = a;
        double r495690 = r495689 * r495685;
        double r495691 = r495688 - r495690;
        double r495692 = r495687 / r495691;
        return r495692;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r495693 = x;
        double r495694 = y;
        double r495695 = z;
        double r495696 = r495694 * r495695;
        double r495697 = r495693 - r495696;
        double r495698 = t;
        double r495699 = a;
        double r495700 = r495699 * r495695;
        double r495701 = r495698 - r495700;
        double r495702 = r495697 / r495701;
        double r495703 = -3.6535243656482265e+306;
        bool r495704 = r495702 <= r495703;
        double r495705 = -7.2576594578206e-315;
        bool r495706 = r495702 <= r495705;
        double r495707 = 0.0;
        bool r495708 = r495702 <= r495707;
        double r495709 = !r495708;
        double r495710 = 5.987964510668617e+295;
        bool r495711 = r495702 <= r495710;
        bool r495712 = r495709 && r495711;
        bool r495713 = r495706 || r495712;
        double r495714 = !r495713;
        bool r495715 = r495704 || r495714;
        double r495716 = r495693 / r495701;
        double r495717 = 3.0;
        double r495718 = pow(r495716, r495717);
        double r495719 = cbrt(r495718);
        double r495720 = r495698 / r495695;
        double r495721 = r495720 - r495699;
        double r495722 = r495694 / r495721;
        double r495723 = r495719 - r495722;
        double r495724 = r495715 ? r495723 : r495702;
        return r495724;
}

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

Original10.4
Target1.8
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x (* y z)) (- t (* a z))) < -3.6535243656482265e+306 or -7.2576594578206e-315 < (/ (- x (* y z)) (- t (* a z))) < 0.0 or 5.987964510668617e+295 < (/ (- x (* y z)) (- t (* a z)))

    1. Initial program 40.9

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

    3. Applied div-sub40.9

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

    4. Simplified25.9

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

    6. Applied *-un-lft-identity25.9

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

    7. Applied associate-*l*25.9

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

    8. Simplified5.8

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

    10. Applied add-cbrt-cube6.4

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

    11. Applied add-cbrt-cube18.7

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

    12. Applied cbrt-undiv18.7

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

    13. Simplified7.5

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

    if -3.6535243656482265e+306 < (/ (- x (* y z)) (- t (* a z))) < -7.2576594578206e-315 or 0.0 < (/ (- x (* y z)) (- t (* a z))) < 5.987964510668617e+295

    1. Initial program 0.2

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

    3. Applied div-sub0.2

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

    4. Simplified1.7

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

    6. Applied associate-*r/0.2

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

    7. Applied sub-div0.2

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

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \le -3.653524365648226522727694876162942276508 \cdot 10^{306} \lor \neg \left(\frac{x - y \cdot z}{t - a \cdot z} \le -7.257659457820581593157072962696615977206 \cdot 10^{-315} \lor \neg \left(\frac{x - y \cdot z}{t - a \cdot z} \le 0.0\right) \land \frac{x - y \cdot z}{t - a \cdot z} \le 5.987964510668616566856227864091206028983 \cdot 10^{295}\right):\\ \;\;\;\;\sqrt[3]{{\left(\frac{x}{t - a \cdot z}\right)}^{3}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))))