Average Error: 10.6 → 3.1
Time: 3.9s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[x \cdot \frac{1}{t - a \cdot z} - \frac{y}{1 \cdot \left(\frac{t}{z} - a\right)}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
x \cdot \frac{1}{t - a \cdot z} - \frac{y}{1 \cdot \left(\frac{t}{z} - a\right)}
double f(double x, double y, double z, double t, double a) {
        double r624830 = x;
        double r624831 = y;
        double r624832 = z;
        double r624833 = r624831 * r624832;
        double r624834 = r624830 - r624833;
        double r624835 = t;
        double r624836 = a;
        double r624837 = r624836 * r624832;
        double r624838 = r624835 - r624837;
        double r624839 = r624834 / r624838;
        return r624839;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r624840 = x;
        double r624841 = 1.0;
        double r624842 = t;
        double r624843 = a;
        double r624844 = z;
        double r624845 = r624843 * r624844;
        double r624846 = r624842 - r624845;
        double r624847 = r624841 / r624846;
        double r624848 = r624840 * r624847;
        double r624849 = y;
        double r624850 = r624842 / r624844;
        double r624851 = r624850 - r624843;
        double r624852 = r624841 * r624851;
        double r624853 = r624849 / r624852;
        double r624854 = r624848 - r624853;
        return r624854;
}

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

Original10.6
Target1.7
Herbie3.1
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.6

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

  3. Applied div-sub10.6

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

  5. Applied associate-/l*8.2

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

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

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

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

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

  9. Applied times-frac8.2

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

  10. Simplified8.2

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

  11. Simplified3.1

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

  13. Applied div-inv3.1

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

  14. Final simplification3.1

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(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))))