Average Error: 10.5 → 1.7
Time: 3.5s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.6144590184121277 \cdot 10^{41}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \le 61778.8119108120518:\\ \;\;\;\;\frac{1}{t - a \cdot z} \cdot \left(x - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x}{t - a \cdot z}} \cdot \sqrt[3]{\frac{x}{t - a \cdot z}}\right) \cdot \sqrt[3]{\frac{x}{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -6.6144590184121277 \cdot 10^{41}:\\
\;\;\;\;\frac{1}{\frac{t - a \cdot z}{x}} - \frac{y}{\frac{t}{z} - a}\\

\mathbf{elif}\;z \le 61778.8119108120518:\\
\;\;\;\;\frac{1}{t - a \cdot z} \cdot \left(x - y \cdot z\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r758008 = x;
        double r758009 = y;
        double r758010 = z;
        double r758011 = r758009 * r758010;
        double r758012 = r758008 - r758011;
        double r758013 = t;
        double r758014 = a;
        double r758015 = r758014 * r758010;
        double r758016 = r758013 - r758015;
        double r758017 = r758012 / r758016;
        return r758017;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r758018 = z;
        double r758019 = -6.614459018412128e+41;
        bool r758020 = r758018 <= r758019;
        double r758021 = 1.0;
        double r758022 = t;
        double r758023 = a;
        double r758024 = r758023 * r758018;
        double r758025 = r758022 - r758024;
        double r758026 = x;
        double r758027 = r758025 / r758026;
        double r758028 = r758021 / r758027;
        double r758029 = y;
        double r758030 = r758022 / r758018;
        double r758031 = r758030 - r758023;
        double r758032 = r758029 / r758031;
        double r758033 = r758028 - r758032;
        double r758034 = 61778.81191081205;
        bool r758035 = r758018 <= r758034;
        double r758036 = r758021 / r758025;
        double r758037 = r758029 * r758018;
        double r758038 = r758026 - r758037;
        double r758039 = r758036 * r758038;
        double r758040 = r758026 / r758025;
        double r758041 = cbrt(r758040);
        double r758042 = r758041 * r758041;
        double r758043 = r758042 * r758041;
        double r758044 = r758043 - r758032;
        double r758045 = r758035 ? r758039 : r758044;
        double r758046 = r758020 ? r758033 : r758045;
        return r758046;
}

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

Original10.5
Target1.6
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -6.614459018412128e+41

    1. Initial program 23.7

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

    3. Applied div-sub23.7

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

    5. Applied associate-/l*14.4

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

    7. Applied div-sub14.4

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

    8. Simplified3.0

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

    10. Applied clear-num3.1

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

    if -6.614459018412128e+41 < z < 61778.81191081205

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    5. Applied div-inv0.5

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

    6. Applied div-inv0.6

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

    7. Applied distribute-rgt-out--0.6

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

    if 61778.81191081205 < z

    1. Initial program 20.9

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

    3. Applied div-sub20.9

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

    5. Applied associate-/l*13.3

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

    7. Applied div-sub13.3

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

    8. Simplified2.8

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

    10. Applied add-cube-cbrt3.1

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.6144590184121277 \cdot 10^{41}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \le 61778.8119108120518:\\ \;\;\;\;\frac{1}{t - a \cdot z} \cdot \left(x - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x}{t - a \cdot z}} \cdot \sqrt[3]{\frac{x}{t - a \cdot z}}\right) \cdot \sqrt[3]{\frac{x}{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))))