Average Error: 10.4 → 2.0
Time: 17.7s
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 r471040 = x;
        double r471041 = y;
        double r471042 = z;
        double r471043 = r471041 * r471042;
        double r471044 = r471040 - r471043;
        double r471045 = t;
        double r471046 = a;
        double r471047 = r471046 * r471042;
        double r471048 = r471045 - r471047;
        double r471049 = r471044 / r471048;
        return r471049;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r471050 = x;
        double r471051 = y;
        double r471052 = z;
        double r471053 = r471051 * r471052;
        double r471054 = r471050 - r471053;
        double r471055 = t;
        double r471056 = a;
        double r471057 = r471056 * r471052;
        double r471058 = r471055 - r471057;
        double r471059 = r471054 / r471058;
        double r471060 = -3.6535243656482265e+306;
        bool r471061 = r471059 <= r471060;
        double r471062 = -7.2576594578206e-315;
        bool r471063 = r471059 <= r471062;
        double r471064 = 0.0;
        bool r471065 = r471059 <= r471064;
        double r471066 = !r471065;
        double r471067 = 5.987964510668617e+295;
        bool r471068 = r471059 <= r471067;
        bool r471069 = r471066 && r471068;
        bool r471070 = r471063 || r471069;
        double r471071 = !r471070;
        bool r471072 = r471061 || r471071;
        double r471073 = r471050 / r471058;
        double r471074 = 3.0;
        double r471075 = pow(r471073, r471074);
        double r471076 = cbrt(r471075);
        double r471077 = r471055 / r471052;
        double r471078 = r471077 - r471056;
        double r471079 = r471051 / r471078;
        double r471080 = r471076 - r471079;
        double r471081 = r471072 ? r471080 : r471059;
        return r471081;
}

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 46.8

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

    3. Applied div-sub46.8

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

    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 pow125.9

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

    7. Applied pow125.9

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

    8. Applied pow-prod-down25.9

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

    9. Simplified3.9

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

    11. Applied add-cbrt-cube4.8

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

    12. Applied add-cbrt-cube20.0

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

    13. Applied cbrt-undiv20.0

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

    14. Simplified6.4

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

    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 2.4

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

    3. Applied div-sub2.4

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

    4. Simplified3.8

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

    6. Applied associate-*r/2.4

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

    7. Applied sub-div2.4

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