Average Error: 10.5 → 2.9
Time: 11.4s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}
double f(double x, double y, double z, double t, double a) {
        double r742055 = x;
        double r742056 = y;
        double r742057 = z;
        double r742058 = r742056 * r742057;
        double r742059 = r742055 - r742058;
        double r742060 = t;
        double r742061 = a;
        double r742062 = r742061 * r742057;
        double r742063 = r742060 - r742062;
        double r742064 = r742059 / r742063;
        return r742064;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r742065 = x;
        double r742066 = 1.0;
        double r742067 = t;
        double r742068 = a;
        double r742069 = z;
        double r742070 = r742068 * r742069;
        double r742071 = r742067 - r742070;
        double r742072 = r742066 / r742071;
        double r742073 = r742065 * r742072;
        double r742074 = y;
        double r742075 = r742067 / r742069;
        double r742076 = r742075 - r742068;
        double r742077 = r742074 / r742076;
        double r742078 = r742073 - r742077;
        return r742078;
}

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
Herbie2.9
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.5

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

  3. Applied div-sub10.5

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

  4. Simplified8.1

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

  6. Applied clear-num8.2

    \[\leadsto \frac{x}{t - a \cdot z} - y \cdot \color{blue}{\frac{1}{\frac{t - a \cdot z}{z}}}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity8.2

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

  9. Applied associate-*l*8.2

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

  10. Simplified2.8

    \[\leadsto \frac{x}{t - a \cdot z} - 1 \cdot \color{blue}{\frac{y}{\frac{t}{z} - a}}\]
  11. Using strategy rm

  12. Applied div-inv2.9

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

  13. Final simplification2.9

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

Reproduce

herbie shell --seed 2020047 
(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))))