Average Error: 10.2 → 1.7
Time: 12.9s
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 r576751 = x;
        double r576752 = y;
        double r576753 = z;
        double r576754 = r576752 * r576753;
        double r576755 = r576751 - r576754;
        double r576756 = t;
        double r576757 = a;
        double r576758 = r576757 * r576753;
        double r576759 = r576756 - r576758;
        double r576760 = r576755 / r576759;
        return r576760;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r576761 = z;
        double r576762 = -1.904673981023646e+40;
        bool r576763 = r576761 <= r576762;
        double r576764 = x;
        double r576765 = t;
        double r576766 = a;
        double r576767 = r576766 * r576761;
        double r576768 = r576765 - r576767;
        double r576769 = r576764 / r576768;
        double r576770 = y;
        double r576771 = 1.0;
        double r576772 = r576765 / r576761;
        double r576773 = r576772 - r576766;
        double r576774 = r576771 / r576773;
        double r576775 = r576770 * r576774;
        double r576776 = r576769 - r576775;
        double r576777 = 1.9458480175732894e-41;
        bool r576778 = r576761 <= r576777;
        double r576779 = r576770 * r576761;
        double r576780 = r576764 - r576779;
        double r576781 = r576780 / r576768;
        double r576782 = r576773 / r576770;
        double r576783 = r576771 / r576782;
        double r576784 = r576769 - r576783;
        double r576785 = r576778 ? r576781 : r576784;
        double r576786 = r576763 ? r576776 : r576785;
        return r576786;
}

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))))