Average Error: 10.2 → 10.4
Time: 20.3s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\left(x - y \cdot z\right) \cdot \frac{1}{t - z \cdot a}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\left(x - y \cdot z\right) \cdot \frac{1}{t - z \cdot a}
double f(double x, double y, double z, double t, double a) {
        double r31347888 = x;
        double r31347889 = y;
        double r31347890 = z;
        double r31347891 = r31347889 * r31347890;
        double r31347892 = r31347888 - r31347891;
        double r31347893 = t;
        double r31347894 = a;
        double r31347895 = r31347894 * r31347890;
        double r31347896 = r31347893 - r31347895;
        double r31347897 = r31347892 / r31347896;
        return r31347897;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r31347898 = x;
        double r31347899 = y;
        double r31347900 = z;
        double r31347901 = r31347899 * r31347900;
        double r31347902 = r31347898 - r31347901;
        double r31347903 = 1.0;
        double r31347904 = t;
        double r31347905 = a;
        double r31347906 = r31347900 * r31347905;
        double r31347907 = r31347904 - r31347906;
        double r31347908 = r31347903 / r31347907;
        double r31347909 = r31347902 * r31347908;
        return r31347909;
}

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
Target1.6
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344.0:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.2

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

  3. Applied *-un-lft-identity10.2

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

  4. Applied associate-/r*10.2

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

  5. Simplified10.2

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

  7. Applied div-inv10.4

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

  8. Final simplification10.4

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))