Average Error: 10.6 → 2.9
Time: 3.6s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - \frac{a}{1}}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - \frac{a}{1}}
double f(double x, double y, double z, double t, double a) {
        double r673879 = x;
        double r673880 = y;
        double r673881 = z;
        double r673882 = r673880 * r673881;
        double r673883 = r673879 - r673882;
        double r673884 = t;
        double r673885 = a;
        double r673886 = r673885 * r673881;
        double r673887 = r673884 - r673886;
        double r673888 = r673883 / r673887;
        return r673888;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r673889 = x;
        double r673890 = 1.0;
        double r673891 = t;
        double r673892 = a;
        double r673893 = z;
        double r673894 = r673892 * r673893;
        double r673895 = r673891 - r673894;
        double r673896 = r673890 / r673895;
        double r673897 = r673889 * r673896;
        double r673898 = y;
        double r673899 = r673891 / r673893;
        double r673900 = r673892 / r673890;
        double r673901 = r673899 - r673900;
        double r673902 = r673898 / r673901;
        double r673903 = r673897 - r673902;
        return r673903;
}

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
Target1.6
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 10.6

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

  3. Applied pow110.6

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

  5. Applied div-sub10.6

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

  6. Simplified8.0

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

  8. Applied div-sub8.0

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

  9. Simplified2.8

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

  11. Applied div-inv2.9

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

  12. Simplified2.9

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

  13. Final simplification2.9

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

Reproduce

herbie shell --seed 2020049 
(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))))