Average Error: 10.2 → 1.7
Time: 12.5s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -19046739810236461113844667691490800041980:\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \le 1.945848017573289371301929976641969901945 \cdot 10^{-41}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{1}{\frac{\frac{t}{z} - a}{y}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -19046739810236461113844667691490800041980:\\
\;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}\\

\mathbf{elif}\;z \le 1.945848017573289371301929976641969901945 \cdot 10^{-41}:\\
\;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r559815 = x;
        double r559816 = y;
        double r559817 = z;
        double r559818 = r559816 * r559817;
        double r559819 = r559815 - r559818;
        double r559820 = t;
        double r559821 = a;
        double r559822 = r559821 * r559817;
        double r559823 = r559820 - r559822;
        double r559824 = r559819 / r559823;
        return r559824;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r559825 = z;
        double r559826 = -1.904673981023646e+40;
        bool r559827 = r559825 <= r559826;
        double r559828 = x;
        double r559829 = t;
        double r559830 = a;
        double r559831 = r559830 * r559825;
        double r559832 = r559829 - r559831;
        double r559833 = r559828 / r559832;
        double r559834 = y;
        double r559835 = 1.0;
        double r559836 = r559829 / r559825;
        double r559837 = r559836 - r559830;
        double r559838 = r559835 / r559837;
        double r559839 = r559834 * r559838;
        double r559840 = r559833 - r559839;
        double r559841 = 1.9458480175732894e-41;
        bool r559842 = r559825 <= r559841;
        double r559843 = r559834 * r559825;
        double r559844 = r559828 - r559843;
        double r559845 = r559844 / r559832;
        double r559846 = r559837 / r559834;
        double r559847 = r559835 / r559846;
        double r559848 = r559833 - r559847;
        double r559849 = r559842 ? r559845 : r559848;
        double r559850 = r559827 ? r559840 : r559849;
        return r559850;
}

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

Original10.2
Target1.7
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.904673981023646e+40

    1. Initial program 23.0

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

    2. Simplified23.0

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

    4. Applied div-sub23.0

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

    5. Simplified14.2

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

    6. Taylor expanded around 0 2.9

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

    8. Applied div-inv3.1

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

    if -1.904673981023646e+40 < z < 1.9458480175732894e-41

    1. Initial program 0.2

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

    2. Simplified0.2

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

    if 1.9458480175732894e-41 < z

    1. Initial program 17.9

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

    2. Simplified17.9

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

    4. Applied div-sub17.9

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

    5. Simplified11.7

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

    6. Taylor expanded around 0 2.9

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

    8. Applied clear-num3.2

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -19046739810236461113844667691490800041980:\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \le 1.945848017573289371301929976641969901945 \cdot 10^{-41}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{1}{\frac{\frac{t}{z} - a}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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))))